Manifolds with Aspherical Singular Riemannian Foliations

Manifolds with aspherical singular Riemannian foliations

Zur Erlangung des akademischen Grades eines DOKTORSDERNATURWISSENSCHAFTEN

von der Fakult¨at fur¨ Mathematik des Karlsruher Institut fur¨ Technologie (KIT)

genehmigte DISSERTATION

von Diego Corro Tapia

Datum der mundlichen¨ Prufung:¨ 4. Juli 2018 Referent: Prof. Dr. Wilderich Tuschmann Korreferent: PD. Dr. Fernando Galaz-Garc´ıa KIT – Universit¨at des Landes Baden-Wurttemberg¨ und nationales Forschungszentrum der Helmholtz-Gesellschaft http://www.kit.edu/

Formated using: The ksp-thesis class. All ﬁgures created using Inkscape and TikZ.

This work is licensed under a Creative Commons “Attribution-ShareAlike 4.0 International” license. Fur¨ meine Eltern A mis padres

ABSTRACT

In the present work we study A-foliations, i.e. singular Riemannian foliations with regular leaf aspherical. The main result is that, for a simply-connected closed (n + 2)- manifold M, an A-foliation with regular leaves of codimension 2 in M is homogeneous. In other words it is given by a smooth eﬀective action of the torus Tn on M by isometries. We will give some conditions to compare two simply-connected, closed manifolds with A-foliations, up to foliated homeomorphism, via their leaf spaces.

Film is one of the three universal languages, the other two: mathematics and music. — Frank Cappra Non est regia ad Geometriam via. — Euclid.

ACKNOWLEDGMENTS

During the research project which culminated in this thesis, I received support from several institutions and people. First of all I would like to thank my advisors Prof. Wilderich Tuschmann, and PD. Dr. Fernando Galaz-Garc´ıafor encouraging me to undertake this project. I am ever grateful to them for the discussions we had, as well as for sharing their knowledge and insight into the fascinating world of geometry. In particular I want to thank Prof. Tuschmann for providing a framework in which I was able to visit several mathematical institutions around the world. This had as a direct consequence, that I was able obtain a very panoramic vision of the field of geometry, as well as the opportunity to interact to several people who helped to enrich this work. To Fernando I am thankful for making a very open and creative work ambient at the institute, which facilitates the discussion and development of ideas. I also want to thank my other colleagues in the Differential Geometry Group at the KIT, Jan-Bernarhd Kordaß, Karla Garc´ıa,and Martin Günter, for the discussions we had on common interests. In particular I want to thank Jan-Bernhard Kordaß for reading the drafts of this work, and sharing his comments, as well as for being an outstanding colleague. I also want to thank Catherina Campagnolo for sharing her knowledge and insight. There are special mentions to Prof. Luis Guijarro and Prof. Thomas Farrell, for pointing to directions that helped me point out the diffeomorphism types of the leaves. I want to thank also Prof. Alexan- der Lytchak for discussions which led to finding obstructions for the existence of a cross-section. During my research program I had the opportunity of visiting the University of Notre Dame. I am grateful to Professor Marco Radeschi, for hosting me and helping me to weed through the technicalities, at a point of the project when time was pressing. I also want to thank Professor Karsten Grove for discussion which led to a better presentation of the subject in this present work. I want to thank Adam Moreno for all the interesting discussions we had about foliations, geometry and life.

iii iv acknowledgments

In particular my I was able to incorporate a different view point of foliations from these discussions. Also thanks to the Institute Henri Poincaréfor providing a very nice library, where I had the pleasure to work for a few weeks. Last I want to thank all the people that lent support in some way during this doctoral research. First of all to Professor Oscar Palmas who suggested me to contact the group of Professor Tuschmann. I am also grateful for the conversations we have every time I go back to Mexico. Thanks to Dr. JesúsNuñes-Zimbrón,Dr. David Gonazález-Alvaro,´ Dr. Masoumeh Zarei, Jaime Santos-Rodr´ıguez,Agustin Romano-Velázquezfor sharing their passion for geometry. To JoséLuis Cisneros for his support and the amazing basis of knowledge with which I started this project. To my friends who supported and encouraged me to keep going on during these last 3 years: Adrián,Iker, David, Luigi, Citlali, Lau, Corinto, Irene, Sean and Gina, and Valerio. I am grateful to my brother Xavier for keeping me humble. Dedications go to Ana Lucia and Gaby for always believing in me. To my dad Leonardo and my mom Ailali for all their love and support. Special thanks to Rebeca for all her love, support, delicious food, and her patience, specially this last six months. The present doctoral work was developed and written under the support of CONACyT–DAAD Scholarship No. 409912.

Karlsruhe, 4. Juni 2018 CONTENTS

Abstracti Acknowledgments iii 1 Introduction1

I Background 11 2 Group Actions 13 2.1 Compact Lie Group Actions ...... 13 2.2 Orbit Types ...... 18 2.3 Torus Actions ...... 21 2.4 Isometric Actions ...... 28 3 Riemannian Foliations 33 3.1 Singular Riemannian foliations...... 33 3.2 Inﬁnitesimal foliation...... 36 3.3 Holonomy and types of leaves ...... 39 3.4 Homogeneous foliations ...... 45

II Aspherical Foliations 49 4 Cross-sections and A-foliations 51 4.1 Cross-section for the leaf space ...... 51 4.2 A-foliations ...... 63 4.3 Molino Bundle ...... 66 4.4 Weights of an A-foliation ...... 71 5 A-foliations of codimension 2 79 5.1 Leaf space of A-foliations of codimension 2 ...... 79 5.2 Weights of A-foliation of codimension 2 ...... 82 5.3 Top. classiﬁcation of A-foliations of codim. 2 ...... 83 6 Smooth Structure of Leaves of an A-Foliation 89 6.1 Fibrations between leaves ...... 90 6.2 Four dimensional torus...... 91 6.3 Higher dimensional torus...... 92

Appendix 99 A Linearized Flows 101 1 Linearized vector ﬁelds ...... 101 ReferencesI

Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions.

— Felix Klein

1 INTRODUCTION

When studying a Riemannian manifold M, an approach to understand its geometry or its topology is to simplify the problem by “reducing” M to a lower dimensional space B. This can be achieved by considering a partition of the original manifold M into submanifolds which are, roughly speaking, compatible with the Riemannian structure of M. We then study the geometry or topology of B, with the aim of recovering information on M.

As an example of this “reduction”, we can consider Riemannian submersions from M onto lower dimensional manifolds. We then study the properties of M which remain invariant along the ﬁbers of the submersion. A concrete example of this is present in [GG87] and [Wil01]. The authors prove that a closed, simply- connected, Riemannian manifold M with sectional curvature greater or equal to 1, and diameter equal to π/2 is either homeomorphic to a sphere, or isometric to a compact symmetric space of rank one (a so called CROSS). As a key step in the proof, they show that any Riemannian submersion π : Sn → B onto some Riemannian manifold B is a Hopf ﬁbration.

This “reduction” approach is also present when we consider Riemannian manifolds with an eﬀective isometric action by a compact Lie group. In particular, this approach has been applied to the long-standing open problem in Riemannian geometry, of classifying and constructing Riemannian manifolds of positive or nonnegative (sectional) curvature. Namely Grove has proposed in the symmetry program to ﬁrst consider such manifolds with a high degree of symmetry, i.e. with an isometric action of a compact Lie group (see [Gro02]).

1 2 introduction

The philosophy behind this approach is that by understanding first positively or nonnegatively curved manifolds with symmetry one may gain insight into the general case, either by constructing new examples or by finding possible obstructions. This has proved a successful approach, since many results have come to light by following loosely the symmetry program (see for example [Bre72], [Gro02], [Gro17], [Kob95], [Sea14],[Wil06]). This point of view has even been applied to other lower curvature bounds, such as positive Ricci (see for example [CGG16]), as it provides many tools and much flexibility.

Since, in particular, any compact connected Lie group contains a maximal torus as a Lie subgroup, the study of torus actions is of importance in the study of group actions. The classiﬁcation up to equivariant diﬀeomorphism of smooth, closed, simply-connected, manifolds with torus actions is a well studied problem when either the dimension of the manifolds or the cohomogeneity of the action is low (see for example [OR70],[KMP74],[Fin77],[Oh83a],[Oh82]).

Both of these phenomena, Riemannian submersions and compact Lie group actions, are encompassed in the more general concept of singular Riemannian foliations. In Riemannian geometry, singular Riemannian foliations have recently attracted the attention of many authors (see, for example, the survey [ABT13]) and led to many interesting results.

Alexandrino has obtained information on the geometry of a manifold admitting certain types of singular Riemannian foliations, called polar foliations (see [Ale10, ABT13]). Singular Riemannian foliations have also led to results in differential topology, such as those surveyed in [QG16]. For example, one can obtain a lower bound on the number of distinct smooth structures a manifold with a singular Riemannian foliation can have. Also, as in the case of smooth effective torus actions, Radeschi and Ge obtained in [GR15] an explicit classification up to diffeomorphism of closed simply-connected 4-manifolds admitting a singular Riemannian foliation. introduction 3

One main diﬀerence between group actions and foliations is that foliations may be less rigid (see for example [GR15]), not having several constraints natural to Lie groups. This in turn raises technical challenges, such as the fact that the leaves may carry non-standard smooth structures.

Thus an important problem in the setting of singular Riemannian foliations is to distinguish homogeneous foliations from non-homogeneous ones (see for example [GR15]). This problem does not become more tractable when the topology of the manifold is not complicated. Even in the case of spheres it is not clear how to distinguish homogeneous foliations (i.e. those coming from group actions), from non-homogeneous ones.

As a concrete example of this, Radeschi studied in his Ph.D. thesis ([Rad12]) singular Riemannian foliations on round spheres, and showed that when the singular foliation has positive dimension at most 3 they are homogeneous. In con- trast when we assume that the foliation has large dimension, for example when the codimension of the singular Riemannian foliation (Sn, F) is 1, non-homogeneous foliations arise. The leaf space of a codimension one foliation (Sn, F) is isometric to the closed interval [0, π/g] with g ∈ {1, 2, 3, 4, 6} (see for example [M¨u80]). For g equal to 1, 2, or 3, Cartan proved that such a foliation is homogeneous (see [Car38, Car39a, Car39b, Car40]), and asked if this was true for all codimension one foliations on Sn. This is answered negatively, when we consider the case g = 4, which includes the majority of codimension 1 singular Riemannian foliations. In [FKM81], for g = 4, an inﬁnite family of non-homogeneous codimension one foliations on round spheres called of FKM type were presented and studied. Following a similar approach, in [Rad14], Radeschi showed the existence of a large family of non-homogeneous singular Riemannian foliations on round spheres with arbitrary codimension. 4 introduction

One of the aims of the present work, is to extend results of the theory of transformation groups to the setting of singular Riemannian foliations. We will focus on the very general problem of comparing two diﬀerent manifolds, each one endowed with a singular Riemannian foliation, via the leaf space, which is the topological space obtained as a quotient of the foliated manifold by the equivalence relation given by the foliation.

The results presented in this work may in turn be applied to the study of Rie- mannian manifolds with positive or nonnegative curvature, generalizing the Grove program to the context of singular Riemannian foliations, as ﬁrst done in [GGR15]. The problem of comparing singular Riemannian foliations encompasses the problem of comparing manifolds with group actions.

To impose some control on this general problem, we impose some control on the topology of the leaves. Namely we focus on compact, simply-connected manifolds with a singular Riemannian foliations with closed aspherical leaves. This means

that for any leaf L of such a foliation πi(L) = 0 for i 6= 1. These type of singular Riemannian foliations are denoted as A-foliations, and they where introduced in [GGR15]. The concept of A-foliations are generalizations of smooth eﬀective torus actions on smooth manifolds.

Galaz-Garc´ıaand Radeschi in [GGR15] give a classification up to foliated diffeomorphism of all compact, simply-connected manifolds with a codimension one A-foliations. They show that they are homogeneous, i.e. these foliations arise from torus actions. They also classify up to homeomorphism all compact, simply- connected, Riemannian manifolds of dimensions 4 and 5 with nonnegative sectional curvature that admit an A-foliation of codimension 2. For the 4-dimension case, from [GR15], it follows that the classification given in [GGR15] is up to diffeomorphism. introduction 5

The main result of the present work is to prove that A-foliations of codimension 2 on compact, simply-connected manifolds are homogeneous up to foliated diﬀeomorphism.

TheoremG. For a compact, simply-connected, Riemannian n-manifold M with n > 4, an A-foliation of codimension 2 is homogeneous.

To be able to prove TheoremG we need to develop a method for comparing two compact, simply-connected manifolds with A-foliations. A technique for classifying up to homeomorphism compact, manifolds admitting a smooth eﬀective Lie group action, is to compare the orbit space (see for example [OR70], [KMP74],[Fin77], [Oh83a]). We would like to apply the same idea to smooth manifolds admitting a singular Riemannian foliation. Let (M1, F1) and (M2, F2) be two compact manifolds admitting singular Riemannian foliations (not necessarily A-foliations). In order to ∗ ∗ be able to compare them by comparing their leaf spaces M1 and M2 , the existence of ∗ ∗ cross-sections σi : Mi → Mi for the quotient map πi : Mi → Mi , (i.e. πi ◦ σi = IdM ∗ ) is extremely useful.

We show the existence of such cross-sections under certain technical topological conditions. We deﬁne the principal stratum of a singular Riemannian foliation (M, F) to be the set of all leaves of maximal dimension with trivial holonomy and denote it by Mprin. The holonomy condition means that a small tubular neighborhood in M of a leaf in Mprin looks like a product of the leaf and a disk (a tube). We also assume that principal leaves are simple. Loosely this means that for a principal k leaf L we have πk(L) = [S , L] for all k > 0. We denote the mapping cylinder of ∗ π : M → M by Mπ. We recall that there is an action of π1(Mprin) on the groups

πk(Mπ, Mprin). When this action is trivial, we call the pair (Mπ, Mprin) simple (this will be precised in section 4.1, or the reader can consult [Hat10, DK01]). With these concepts we ﬁnd a family of obstructions to the existence of a cross-section over

Mprin. 6 introduction

TheoremA. Let (M, F) be a closed singular Riemannian foliation with M simply- connected, quotient map π : M → M ∗, and principal leaf L, which is simple and ∗ connected. Furthermore assume Mprin is simply-connected and (Mπ, Mprin) is simple. 1 k+1 ∗ Then there is a family of obstructions ωk ∈ H (Mprin; πk(L)) such that a cross- ∗ 1 section σ : Mprin → Mprin exists if ωk = 0 for all k.

∗ Next we consider the problem of extending a cross-section σ : Mprin → Mprin to the whole orbit space M ∗. We show the existence of a second family of obstructions to the extension problem.

TheoremB. Let (M, F) be a closed singular Riemannian foliation with M simply connected, and consider the quotient map π : M → M ∗. Furthermore assume ∗ that the homotopy ﬁber Fπ is simple, and setting A = Mprin assume there is al-

ready a deﬁned cross-section σ : A → Mprin. Then there is a family of obstructions 2 k+1 ∗ ∗ ωk ∈ H (M , A; πk(Fπ)) such that a cross-section σ˜ : M → M extending σ exists 2 if ωk = 0 for all k.

From these two theorems we get the following corollary, which gives a suﬃcient condition for the existence of a cross-section.

CorollaryC. Let (M, F) be a closed singular Riemannian foliation on a simply- ∗ connected manifold. Suppose that there is a section σ˜ : Mprin → Mprin, and the that ∗ ∗ hypotheses of TheoremB are satisﬁed. If Mprin has the same homotopy type of M , then the cross-section σ˜ can be extended to a section σ.

We then proceed to study A-foliations on compact, simply-connected manifolds, namely the homeomorphism type of the leaves. We will prove that, except for possibly dimension 4, they are all homeomorphic to tori, or Bieberbach manifolds. This is due to the positive answer to the Borel conjecture for virtually Abelian groups, except in dimension 4 (see for example [FH83],[KL09]). introduction 7

Conjecture (Borel conjecture). Given two aspherical closed topological manifolds X and Y , and f : X → Y a homotopy equivalence. Then f is homotopic to an homeomorphism.

We also study the infinitesimal foliations of A-foliations, as well as the holonomy of the leaves, and propose a finer stratification of the manifold. Both of these concepts in the particular case of homogeneous foliations are encoded in the isotropy of a leaf. We define the weights of the foliation, which encode the information of the infinitesimal foliation and the holonomy. They generalize the weights of smooth effective torus actions (defined in [OR70], [Fin77], [Oh83a]), which encode the isotropy information of torus actions. In the case of existence of a cross-section, the weights characterize up to foliated homeomorphism the manifold.

TheoremD. If (M1, F1) and (M2, F2) are compact, simply connected manifolds, with A-foliations, such that they have isomorphic weighted leaf spaces and admit ∗ cross-sections σi : Mi → Mi, then (M1, F1) is foliated homeomorphic to (M2, F2).

In the general setting of classifying manifolds with singular Riemannian foliations via cross-sections, the best one can obtain is a classiﬁcation up to foliated homeomorphism. This is because the leaf spaces are only metric spaces (i.e. they may not even be topological manifolds). In the case of A-foliations of codimension 2 on compact, simply-connected manifolds, the authors in [GGR15] proved that the leaf space is homeomorphic to a 2-dimensional disk D2. For this case the boundary points of the leaf space correspond exactly to the singular leaves of F. Since this leaf space satisﬁes the conditions of CorollaryC, it follows from TheoremD that compact, simply-connected manifolds with A-foliations of codimension 2 are characterized up to foliated homeomorphism by the weights of the foliations.

TheoremE. Let (M1, F1) and (M2, F2) be two compact, simply connected smooth (n + 2)-manifolds, admitting singular A-foliations of codimension 2 and n > 2. 8 introduction

∗ Then M1 is foliated homeomorphic to M2 if and only if the weighted leaf spaces M1 ∗ and M2 are isomorphic.

Furthermore Oh in [Oh83a], shows that given a weighted 2-disk, with the weights satisfying some conditions, there is a procedure to construct a smooth manifold with an eﬀective smooth action of cohomogeneity two realizing the weighted disk as an orbit space.

Theorem 2.22[Oh83a]. For n > 2 and a family of legal weights (ai1, ... , ain) ∈ Zn there exists a closed, simply-connected (n + 2)-manifold admitting a cohomo- n geneity two T -action, that realizes the family (ai1, ... , ain) as weights.

We will show that the weights of an A-foliation of codimension two on a compact, simply-connected manifold M are legal weights in the sense of Oh. Thus there is a torus action on M with the same weights as the foliation. By TheoremD we conclude that an A-foliation of codimension two on a compact, simply-connected manifold is, up to foliated homeomorphism, a homogeneous foliation.

TheoremF. Let (M, F1) be a compact, simply connected (n + 2)-manifold with an A-foliation of codimension 2 and n > 2. Then there exists a closed, simply-

connected (n + 2)-manifold (N, F2) with a homogeneous A-foliation of codimension

2 (i.e. with an eﬀective smooth torus action of cohomogeneity 2), such that (N, F2)

is foliated homeomorphic to (M, F1).

As mentioned before in the problem of classifying manifolds with singular Rie- mannian foliations via cross-sections, the best one can obtain is a classification up to foliated homeomorphism. But in the case of A-foliations of codimension two on compact, simply-connected spaces, the leaf space is a 2-disk (see [GGR15]), and thus it is a smooth manifold with boundary in a unique way (it admits a unique smooth structure). So we can expect, in this case, to get a classification up to foliated diffeomorphism. The next obstruction is the existence of exotic smooth structures on introduction 9 torus (see for example [HS69], [HS70]). We study the diffeomorphism type of the leaves of an A-foliation of codimension two on a simply-connected smooth manifold, and prove that they are diffeomorphic to standard tori. With this we conclude the main theorem of this work:

TheoremG. For n > 2, every A-foliation of codimension 2 on a compact, simply- connected Riemannian (n + 2)-manifold M is homogeneous.

We state the presentation order of the present work. In the first part of this work we will give an overview of the theory of Lie group actions and singular Riemannian foliations. In the second part, we will focus on the study of cross-sections and A- foliations on compact, simply-connected manifolds. In section 4.1 we give proofs for TheoremA, TheoremB and CorollaryC. In section 4.4 we define the weights of an A-foliation and prove TheoremD. In chapter5 we focus on the study of A- foliations of codimension 2 on simply-connected manifolds, and prove TheoremE, and TheoremF. Finally on chapter6 we study the diffeomorphism type of the leaves of an A-foliation of codimension 2 on a simply-connected manifold, and finish the proof of TheoremG.

Part I

BACKGROUND

2 GROUPACTIONS

In this chapter we review the theory of compact Lie group actions on smooth manifolds, stating some basic results, and then explain some results of Orlik, Raymond and Oh in [OR70, Oh83a]. For a more comprehensive presentation of the subject, the interested reader can consult [AB15, Bre72]

2.1 compact lie group actions

Let us begin by stating the basic concepts and results for diﬀerentiable group actions.

If M is a smooth manifold and G is a Lie group with identity element e, a smooth group action of G on M is a smooth map

µ: G × M → M, such that for any elements g, h of G and any element x in M the following hold:

(i) µ(e, x) = x,

(ii) µ(gh, x) = µ(g, µ(h, x)).

By setting µ(g, x) = x for any g ∈ G we see that there always exists a group action. This action is called the trivial action. From now on we will use the following

13 14 group actions

more compact and generally used notation for group actions g(x) := µ(g, x). This actually defines a representation of G into Diff(M), the diffeomorphism group of M. Thus we may consider the differential Dg of an element g ∈ G as the differential of

the map µg : {g} × M → M.

The kernel of the action µ is the closed normal subgroup

ker µ = {g ∈ G | g(x) = x for all x ∈ M}.

The action µ is called effective if ker µ is trivial. From the following proposition we see that from now on we may consider only effective group actions. Proposition 2.1 (Proposition 1.1, Chapter I in [Bre72]). Let µ be an action of G on M and set N = ker µ. Then there is a canonically induced effective action µ/ ker µ of G/N on M.

When working with compact Lie groups G one has, for the map µ, the following nice properties. Theorem 2.2 (Theorem 1.2, Chapter I in [Bre72]). If µ: G × M → M is an action of a compact Lie group G on M, then µ is a closed map. Corollary 2.3. If G is a compact Lie group acting on M, then G(A) is closed (compact) in M for each closed (compact) A ⊂ M.

An action µ: G × M → M is proper if the map ϕ: G × M → M × M given by

ϕ(g, x) = (g(x), x),

is proper. We have the following characterization for proper actions. Proposition 2.4 (Proposition 3.19 in [AB15]). An action µ: G × M → M is proper

if and only if for any sequence {gn} in G and any convergent sequence {xn} in M,

such that {gn(xn)} converges, the sequence {gn} has a convergent subsequence 2.1 compact lie group actions 15

Proof. Suppose that the action is proper and {gn} is a sequence in G, {xn} is a sequence converging in M to x, such that {gn(xn)} is a convergent sequence in

M with limit y. Then the sequence {(gn(xn), xn)} is convergent in M × M, with limit (y, x). The set K = {(gn(xn), xn)} ∪ {(y, x)} is a compact subset of M × M, −1 −1 and thus ϕ (K) is compact in G × M. Since we have {gn, xn} ⊂ ϕ (K) there is a convergent subsequence of {gn, xn} in G × M and thus there is a convergent subsequence of {gn} in G.

Conversely we assume that for any sequence {gn} in G and any convergent sequence {xn} in M, such that {gn(xn)} converges, the sequence {gn} has a convergent subsequence. Take K ⊂ M × M compact. Now consider a sequence −1 {gn, xn} in ϕ (K). Then the sequence {gn(xn), xn} has a convergent subsequence

{(gn (xn ), xn )} in K. Thus there is a convergent subsequence {gn } of {gn }. k k k ki k −1 Thus we have showed that for any sequence {gn, xn} in ϕ (K) there is a convergent subsequence {(gn , xn )}. ki ki Corollary 2.5. For an action µ: G × M → M, if G is a compact group, then the action is proper.

Proof. Every sequence {gn} of G has a convergent subsequence in G.

Now we consider two smooth manifolds M and N that have a group action µ: G × M → M and θ : G × N → N. We say that a smooth function f : M → N is equivariant if for all p ∈ M and all g ∈ G the following is true:

f(g(x)) = g(f(x)).

If the equivariant smooth function f : M → N is also a diﬀeomorphism then we say that the action µ of G on M is equivalent to the action θ of G on N, and we say that M is equivariantly diﬀeomorphic to N. 16 group actions

The following two concepts are at the core of the study of group actions. First, for a point p ∈ M, we deﬁne the orbit of p under G as

G(p) = {g(p) | g ∈ G}.

We also deﬁne the isotropy subgroup of G at p as

Gp = {g ∈ G | g(p) = p},

−1 i.e. the subgroup of all elements in G which ﬁx p. Since gGpg (g(p)) = g(p) we −1 −1 have that gGpg ⊂ Gg(p), and conversely since g Gg(p)g(p) = p we have,

−1 Gg(p) = gGpg . (2.1.1)

Thus for a given point p ∈ M its isotropy subgroup Gp changes by conjugation as the point p moves along its orbit G(p). It is easy to prove that if two orbits G(p) and G(q) have non-empty intersection, then they coincide, and thus the orbits of the action of G on M form a partition of M. Proposition 2.6. For a smooth Lie group action µ: G × M → M, the orbits of µ give a partition of M.

Proof. Clearly since p ∈ G(p) for any p ∈ M, no orbit is empty, and M = ∪p∈M G(p). Let q, p ∈ M be such that G(p) ∩ G(q) 6= ∅. Then there exists x ∈ G(p) ∩ G(q).

Since x is an element in G(p) we have by deﬁnition that x = g1(p) for some g1 ∈ G. −1 −1 Analogously we have that x = g2(q) for some g2 ∈ G. Thus q = g2 (x) = g2 g1(p), and therefore q is an element of G(p). From this it follows that G(q) ⊂ G(p). In a similar fashion, interchanging the role of p and q we prove that G(p) ⊂ G(q) and thus we obtain that G(p) = G(q). Thus we have proven that if we have two non-equal orbits, their intersection must be empty. 2.1 compact lie group actions 17

Hence we can consider the quotient given by the equivalence relation induced by the partition, namely,

M ∗ = M/G = {G(p) | p ∈ M}, which is called the orbit space of the action. The natural projection π : M → M/G, given by π(p) = G(p), is called the quotient map. We set a topology on M/G by declaring that U ⊂ M/G is open if and only if its preimage π−1(U) ⊂ M is open, i.e. the quotient topology. This implies that π is continuous and open. For compact groups G, the orbit space has reasonable properties. Theorem 2.7 (Theorem 3.1 in [Bre72]). If a compact group G acts on M, then

(i) M ∗ is Hausdorﬀ.

(ii) π : M → M ∗ is closed.

(iii) π : M → M ∗ is proper.

(iv) M is compact if and only if M ∗ is compact.

(v) M is locally compact if and only if M ∗ is locally compact.

The following proposition shows that for a proper action by a smooth Lie group the orbit of a point is an embedded submanifold. Proposition 2.8 (Proposition 3.41 in [AB15]). Let µ: G × M → M be a smooth

Lie group action and x ∈ M. Deﬁne αx : G/Gx → M as αx(gGx) = g(x). Then αx is a G-equivariant injective immersion with image G(x). If in addition the action is proper, then αx is an embedding, and G(x) is an embedded submanifold of M. 18 group actions

2.2 orbit types

In this section we compare the orbits of an action by comparing the isotropy subgroups. We say that an orbit G(x) is a principal orbit if there exists a neighborhood

V of x in M such that for every y ∈ V there exists some g ∈ G such that Gx ⊂ Gg(y). This deﬁnition does not depend on the choice of the representative of the orbit G(x) since we have already showed that isotropy groups are conjugate along orbits. The principal orbits are the ones that have the smallest isotropy subgroup among nearby orbits. The existence of principal orbits is guaranteed by [Bre72, IV Theorem 3.1] and [AB15, Theorem 3.82] (see Theorem 2.9).

If G(x) = G/Gx is a principal orbit and G(y) = G/Gy is another orbit then, by

deﬁnition, we have that Gx is conjugate to a subgroup of Gy, and so we may assume

without loss of generality that Gx ⊂ Gy. We consider the map p: G(x) → G(y), given by p(g(x)) = g(y). This map is an equivariant surjection, and in fact it

is a ﬁber bundle projection with ﬁber Gy/Gx. If G(x) is a principal orbit, we say that G(y) is a exceptional orbit if dim(G(x)) = dim(G(y)), but they are

not equivalent, meaning that Gy/Gx is a ﬁnite nontrivial group. In case that dim(G(y)) < dim(G(x)) we say that G(y) is a singular orbit. We denote here as P the set of all principal orbits, E the set of all exceptional orbits and Q the set of all the singular orbits.

We can deﬁne a relation between diﬀerent isotropy subgroups. If H is an isotropy

group of G we say that an orbit G(x) has type (H) if Gx is conjugate to H in G. Since along an orbit the isotropy subgroups are conjugated, the type is well deﬁned. For K, another isotropy subgroup of G, we say that the orbit type (H) is less than or equal to the type of orbit (K) and we denote it by (H) ≺ (K), if K is conjugate to a subgroup of H in G. With this relation we see that if G(x) is a principal orbit, then there exists an open neighborhood V of x in M such that for every y ∈ V we 2.2 orbit types 19

have that (Gy) ≺ (Gx). We say that two orbits G(x) and G(y) have the same orbit type if (Gx) ≺ (Gy) and (Gy) ≺ (Gx). With the following theorem we ensure the existence of principal orbits, and that there is only one principal orbit type

Theorem 2.9 (Principal Orbit Theorem 3.82 in [AB15]). Denote by Mprinc the set of points contained in principal orbits. Then the following hold:

(i) Mprinc is open and dense in M.

(ii) The subset Mprinc/G of M/G is a connected manifold.

(iii) If G(x) and G(y) are principal orbits, then there exists g ∈ G such that

Gx = Gg(y).

Now we look at the decomposition of the tangent space of the orbit G(p) at the point p. We say that an embedded submanifold Sp of M containing p is a slice at p if it satisﬁes the following properties:

(i) TpM = TpG(x) ⊕ TpSp.

(ii) TxM = TxG(x) + TxSp for all x ∈ Sp.

(iii) Sp is invariant under the action of Gp, i.e. if x ∈ Sp and g ∈ Gp , then

g(x) ∈ Sp.

(iv) If x ∈ Sp and g ∈ G are such that g(x) ∈ Sp, then g ∈ Gp.

Thus we have a way to decompose at each point p the tangent space TpM. We will refer to TpSp as the normal space of G(p) at p. The existence of a slice is guaranteed by the following theorem. Theorem 2.10 (Slice Theorem, Theorem 3.49 in [AB15]). For any compact (proper) group action µ: G × M → M there exists a slice Sx0 at x0 for any x0 ∈ M. 20 group actions

Let µ: G × M → M be a proper smooth action, for x0 ∈ M ﬁxed, we deﬁne a

tubular neighborhood of the orbit G(x0) to be the image of Sx0 , the slice through x0, under the G-action:

Tub(G(x0)) = µ(G, Sx0 ).

By the following theorem the tubular neighborhood Tub(G(x0)) is the total space of a ﬁber bundle. Theorem 2.11 (Tubular Neighborhood Theorem, Theorem 3.57 in [AB15]). Let

µ: G × M → M be a smooth proper action. For every point x0 in M there exist a G-

equivariant diﬀeomorphism between Tub(G(x0)) and the total space of the Sx0 -ﬁber bundle with,

Sx0 → G ×H Sx0 → G/H,

associated to the principal bundle H → G → G/H. Here H = Gx0 is the isotropy

subgroup at x0.

If we now consider x ∈ M we can get an invariant tubular neighborhood around

the orbit G(x), considering the slice Sx through x and setting

Tub(G(x)) = G(Sx).

Furthermore in the tubular neighborhood we will have a ﬁnite number of orbit types. Theorem 2.12 (Theorem 3.91 in [AB15]). For a compact group action, for every

x ∈ M there exists a slice Sx at x, such that the tubular neighborhood G(Sx) contains only ﬁnitely many diﬀerent orbit types.

Thus when the manifold M is compact, since we can cover it by a finite number of such tubular neighborhoods, we conclude that the number of orbit types is finite. Corollary 2.13. If M is a compact manifold and G is a compact group acting on M, then the number of orbit types is finite. 2.3 torus actions 21

For a compact group action µ: G × M → M we have that for each g ∈ G we can define a diffeomorphism µg of M as µg(x) = g(x). Now fix a point p ∈ M and consider g ∈ Gp, so that µg fixes p. Thus for each point p ∈ M we can define an action µ˜ : Gp × TpM → TpM, by setting for g ∈ Gp and v ∈ TpM the function µ˜ as follows,

µ˜(g, v) = Dp(µg)(v), where Dp(µg) is the derivative of µg at p. If we consider the slice Sp at p, since it is invariant under the action of Gp, we have that Gp acts on the normal space of G(p) at p a via µ˜. This action is called the isotropy representation of the action at p.

We end this section by reviewing the following concept. For an action of G over an n-dimensional manifold M we deﬁne the cohomogeneity as the codimension of a principal orbit (which has maximal dimension). If the cohomogeneity is small enough we have the following lemma that helps us to understand the orbit space M ∗. Lemma 2.14 (Chapter IV, Lemma 4.1 in [Bre72]). For a compact group action G over M, if the cohomogeneity is less than or equal to 2 then the orbit space M ∗ is a manifold with boundary of dimension equal to the cohomogeneity.

From now on we will be working with cohomogeneity 2 group actions.

2.3 torus actions

Since every compact Lie group G contains a unique maximal torus, torus actions play an important role in the study of compact group actions. In this section we will concentrate on the case where the group G acting eﬀectively is an n-torus and the (n + 2)-manifold M is simply connected and closed. Following the notation 22 group actions

of [BFJ16], we will denote the standard n-torus by Tn, and the circle group by T1, to distinguish it from the circle S1 as topological space (i.e. without a group structure). Proposition 2.15. An eﬀective action of Tn on a simply-connected (n + 2)-manifold has cohomogeneity 2.

Proof. The trivial subgroup {e} ⊂ Tn is an isotropy subgroup of the action for some

point p ∈ M, and since {e} ⊂ Gx for any point x, we have that the orbit is principal. Thus we have from 2.8 that Tn(P ) is diﬀeomorphic to Tn and therefore the action has cohomogeneity 2.

In general for an action of cohomogeneity 2, the following theorem tells us the orbit space structure. Theorem 2.16 (Chapter IV, Theorem 8.6 in [Bre72]). If G is a connected Lie group acting on a compact simply-connected manifold M with cohomogeneity 2 and there exists a singular orbit, then the set of exceptional orbits is empty and the orbit space M ∗ is a 2-disk whit boundary Q∗.

Furthermore from [KMP74, Theorem 1.3] and [GGK14] the only possible non- trivial isotropy subgroups of an eﬀective action of Tn on M, are T1 and T2. The ∗ boundary circle Q is a union of m > n arcs by [KMP74, Corollary 1.7]. The interior points of the arcs corresponds to orbits with isotropy T1 and the end points correspond to orbits with isotropy T2, as shown in Figure 2.1.

Now considering the torus Tn = Rn/Zn as the quotient of Rn by an integer n lattice, we deﬁne a circle subgroup G(a1, ... , an) of T as a the projection under n n n R → T of a line in the direction of (a1, ... , an) ∈ Z . The vectors (1, 0, ... , 0) and (2, 0, ... , 0) represent the same circle subgroup in Tn, so in order to represent

uniquely the possible circle subgroups, the integers a1, a2, ... , an must be relatively prime. 2.3 torus actions 23

T2 isotropy M ∗ Trivial isotropy

T1 isotropy

Figure 2.1.: Orbit space structure of a cohomogeneity-two torus action on a closed, simply- connected manifold.

By the determinant of n-circle subgroups

G(a11, a12, ... , a1n), G(a21, a22, ... , a2n), ... , G(an1, an2, ... , ann), of T n we mean the determinant of the matrix:

  a11 a12 ··· a1n       a21 a22 ··· a2n   .  . . . .   . . .. .      an1 an2 ··· ann

The determinant of n-circles subgroups characterizes the intersection of two circle subgroups, as seen in the following lemma.

Lemma 2.17 (Lemma 1.2 in [Oh83a]). Two circle subgroups G(a1, ... , an) and n G(b1, ... , bn) of T have trivial intersection if and only if there exist n − 3 vectors n Gi ∈ Z with i = 3, ... , n, such that the determinant of (a1, ... , an), (b1, ... , bn),

, G3, ... , Gn is ±1. 24 group actions

Furthermore the determinant of a family of integer vectors

n {(a11, ... , a1n), (a21, ... , a2n), ... , (an1, ... , ann)} ⊂ Z

tells us when do they generate a torus.

Lemma 2.18 (Lemma 1.4 in [Oh83a]). The n-circles G(a11, ... , a1n), n , G(a21, ... , a2n), ... , G(an1, ... , ann) generate T if and only if the determinant of the n circles is ±1.

∗ We order the edges of Q and label them by γ1, ... , γm. Next we show that for

two orbits G(x) and G(y) that project to the same edge γi under π : M → M/G,

they have the same type G(ai1, ... , ain). We also now that for a vertex Fi the 2 circle subgroups of γi and γi+1 are subgroups of the isotropy subgroup T at Fi (see [KMP74, Theorem 1.3]).

The following proposition tells us that for a Tn-action of cohomogeneity two, there exist at least n circle subgroups, which are isotropy groups of some orbits, and they generate the n-torus. Proposition 2.19 (Corollary 1.7 in [KMP74]). If Tn acts eﬀectively on a simply- connected closed (n + 2)-manifold M, then all isotropy subgroups generate the whole group Tn and there are at least n diﬀerent circle isotropy subgroups of Tn.

Combining Lemma 2.17 with Proposition 2.19 we see that actually for the ver- 2 tex Fi the isotropy subgroup T is generated by the isotropy circle subgroups

G(ai1, ... , ain) and G(ai+11, ... , ai+1n) associated to γi and γi+1 respectively. Thus the circle subgroups carry all the isotropy information of the action. We deﬁne the weights of orbit space M ∗ for the action of Tn as the isotropy circle subgroups

G(ai1, ... , ain). These weights have a crucial role in solving the problem of classifying the smooth simply-connected closed (n + 2)-manifolds M that admit an effective action by Tn. The following theorems show that the action is classified by the weights previously defined. 2.3 torus actions 25

Theorem 2.20 ([OR70, KMP74, Oh83a]). For an eﬀective Tn action on a simply- connected closed (n + 2)-manifold M, the orbit map π : M → M ∗ has a cross-section.

We will give an alternate proof of the previous theorem in Section 4.1. Using the existence of cross-sections we can prove the following theorem, which implies that cohomogeneity-two torus actions on simply-connected, closed, smooth manifolds are classified by the weights. A proof of this theorem will also be given in Section 4.1. Theorem 2.21 ([OR70, KMP74, Oh83a]). Let Tn act effectively on the simply- connected closed (n + 2)-manifolds M and N. There is an equivariant diffeomorphism f of M onto N if and only if there is a weight preserving diffeomorphism f ∗ of M ∗ onto N ∗.

Now consider the disk D2 and split its circle boundary into m edges, and we order n them. If we attach to each edge an n-tuple (ai1, ... , ain) ∈ Z with 2 gcd(ai1, ... , ain) = 1, we say that D is legally weighted with weights

(a11, ... , a1n), (a21, ... , a2n), ... , (am1, ... , amn),

if any two adjacent vectors (ai1, ... , ain) and (ai+11, ... , ai+1n) have trivial intersection in the sense of lemma 2.17. Oh showed in [Oh83a] that given a legally weighted disk, there exists a simply-connected closed (n + 2) manifold M and an eﬀective action of Tn on M that has as weighted orbit space M ∗ the weighted disk we started with. This means that having legal weights is a suﬃcient and necessary condition to characterize simply-connected, compact, (n + 2)-manifolds admitting a Tn-action.

Theorem 2.22 (Remark 4.7 in [Oh83a]). For n > 2 and a family of legal weights n (ai1, ... , ain) ∈ Z there exists a closed, simply-connected (n + 2)-manifold ad- n mitting a cohomogeneity two T -action that realizes the family (ai1, ... , ain) as weights. 26 group actions

We end this section and this chapter presenting the topological classification for simply-connected closed (n + 2)-manifold M that admit an effective Tn action. Theorem 2.23. If Tn acts effectively on a simply-connected closed (n + 2)-manifolds M then the following hold:

(i)[OR70] If n = 2, then M is a connected sum of S4, ±CP 2 and S2 × S2.

2 If there are k orbits with isotropy subgroup T , and w2(M) denotes the second Stiefel-Whitney class of M, we have that

(ii)[Oh83a] If n = 3, then k > 3 and M is diﬀeomorphic to

• S5 if k = 3.

3 2 • #(k − 3)(S × S ) if w2(M) = 0.

3 2 3 2 • (S ×˜ S )#(k − 4)(S × S ) if w2(M) 6= 0,

where S3×˜ S2 is the nontrivial S3 bundle over S2.

(iii)[Oh82] If n = 4, then k > 4 and we have that M is diﬀeomorphic to

4 2 3 3 • #(k − 4)(S × S )#(k − 3)(S × S ) if w2(M) = 0.

4 2 5 2 3 3 • (S ×˜ S )#(k − 5)(S × S )#(k − 3)(S × S ) if w2(M) 6= 0,

where S4×˜ S2 is the nontrivial S4 bundle over S2.

In the case of 4-dimensional manifolds the classification of diffeomorphism type is obtained as follows. First, the authors show that there are 7-basic pieces into which any legally weighted disk can be decomposed. This decomposition may not be unique, with each piece corresponding to one of the basic configurations. Next it is shown that for each basic piece there exists a unique diffeomorphism type of a 4-dimensional, closed, simply-connected manifold with a torus action, whose orbit space is the given basic piece. Namely to each basic piece, the smooth manifold with a torus action realizing this piece as an orbit space is one of S4, ±CP 2, S3 × S2 or 2.3 torus actions 27

S2×˜ S2 (see [OR70, Table at pp. 552]). Last it is proved that the decomposition of the orbit space, given by the pieces, corresponds to a connected sum decomposition of the original manifold, by the previous list of manifolds (see [OR70, Theorem 5.7]). It is also observed that the connected sum presentation is not unique, i.e. it is not equivariant (see [OR70, Remark 5.8]).

For the case of 5-dimensional manifolds, the classification of diffeomorphism type is achieved by showing that the number of orbits with circle isotropy is the rank of the second homology group (see [Oh83a, Lemma 5.4]), and explicitly constructing two families of closed, simply-connected 5-manifolds with a T3-action, realizing all the second homology groups. In the first one, all manifolds have non- trivial second Stiefel-Whitney class, while in the second one they have trivial second Stiefel-Whitney class. The classification follows then from the work of Barden- Smale, in which they show that the second homology group and the second Stiefel- Whitney class classify closed, simply-connected 5-manifolds up to diffeomorphism (see [Bar65, Sma62] and [Oh83a, Theorem 5.5]).

The 6-dimensional case is done as follows. As in the 4-dimensional case, we can show the existence of basic closed, simply-connected, 6-manifolds which determine basic legally weighted orbit spaces. Then we prove that any legally weighted orbit space is obtained via an inductive process from these basic pieces. By computing the homology groups, the ﬁrst Pontryagin classes, and a trilinear form associated to the manifolds obtained via the inductive process, we can apply classiﬁcation theorems by Wall and Jupp to obtain the explicit list (see [Jup73, Wal66]). 28 group actions

2.4 isometric actions

For a Riemannian manifold (M, g) we say that a Lie group action µ: G × M → M is an action by isometries, or an isometric action, if for each h ∈ G the diﬀeomorphisms

µh are isometries, i.e. if for any X, Y ∈ TM the following occurs:

g(X, Y ) = g(Dµh(X), Dµh(Y )).

In this case we also say that the Riemannian metric g is G-invariant.

Recall that for any metric g on a smooth manifold M since Isom(M, g) is a subset of C∞(M, M), the set of all smooth functions from M to M, by endowing C∞(M, M) with the open-compact topology, we can give a topology to Isom(M, g), which we also call the open-compact topology. With this topology on Isom(M, g) Myers and Steenrod showed that the isometry group of a Riemannian manifold is a Lie group. Theorem 2.24 ([MS39]). Let (M, g) be a Riemannian manifold. Any closed subgroup of Isom(M, g) with the compact-open topology is a Lie group. In particular, Isom(M, g) is a Lie group.

Furthermore they show that in the particular case when M is compact, the group Isom(M, g) is a compact Lie group. This implies, via the following theorem, that Isom(M, g) always contains a torus as a subgroup. Theorem 2.25 (Maximal Torus Theorem, Theorem 4.1 in [AB15]). Let G be a connected, compact Lie group. Then the following hold:

(i) There exists a maximal torus T in G;

(ii) Any two maximal tori in G are conjugate;

(iii) Every element of G is contained in a maximal torus. 2.4 isometric actions 29

Furthermore given a smooth group action µ of G on M, the following result shows that there exists a metric under which the given action µ is an isometric action. Theorem 2.26 (Theorem 3.65 in [AB15]). Given a proper smooth action µ: G × M → M, there exists a G-invariant metric g on M such that G is a closed subgroup of Isom(M, g).

Isometric actions have been successfully used to characterize Riemannian manifolds admitting nonnegative or sectional positive curvature as proposed by the so- called Grove Program (see [Gro02]). The general idea is the following: assume (M, g) is a Riemannian manifold with nonnegative (positive) curvature, and assume that a given Lie group G is contained in the isometry group Isom(M, g). In other words we assume that we have a smooth faithful representation ρ: G → Isom(M, g). From this we can use several results from smooth group actions to characterize the diﬀeo- morphism type of M. For example in dimension 4 Hsiang and Kleiner showed that if M is a simply-connected, 4-dimensional Riemannian manifold of positive sectional curvature, and the circle T 1 acts by isometries, then M is homeomorphic to either S4 or CP 4. This can in fact be improved to diﬀeomorphism (see [GW14]).

Since in particular for a compact Riemannian manifold (M, g), as observed above, the group Isom(M, g) contains a torus as a closed subgroup, we may consider this torus as the subgroup G and try to deduce some properties about M. As an example in this direction, Galaz-Garc´ıaand Searle showed in [GGS14] the following. Theorem 2.27 (Theorem A in [GGS14]). Let M be a closed, simply connected, nonnegatively curved 5-manifold. If T 2 acts isometrically and eﬀectively on M, then M is diﬀeomorphic to one of S5, S3 × S2, S3×˜ S2, or the Wu manifold SU(3)/SO(3).

We observe that this approach may be somewhat limited since, by the work of Ebin, for a ﬁxed smooth manifold M most Riemannian metrics g on M have trivial isometry group (see [Ebi70]). To understand how constrained this approach may 30 group actions

be, any type of answer (either positive or negative) to the following problem is required. Problem 2.28 (Problem 5.5 in [Gro02]). Do simply-connected manifolds of nonnegative or more generally almost nonnegative curvature have positive symmetry degree?

In another direction, there is the following question: Given a smooth group action G on a smooth manifold M, does a G-invariant Riemannian metric g exists on M, which admits given lower curvature bounds?

For actions with low cohomogeneity several positive answers have been given, in particular for positive Ricci curvature. For example in [GZ02] the following theorem was proved for cohomogeneity one actions. Theorem 2.29. A compact cohomogeneity one manifold admits an invariant metric with positive Ricci curvature if and only if its fundamental group is ﬁnite.

For a torus action of cohomogeneity two, on a simply-connected, closed, smooth manifold the following was proved in [CGG16]. Theorem 2.30. If M is a closed, simply-connected smooth (n + 2)-manifold with a smooth, eﬀective action of a torus T n, then there exists a T n-invariant Riemannian metric on M with positive Ricci curvature.

In particular from the previous theorem it follows that all spaces in Theorem 2.23 admit an invariant metric with positive Ricci curvature. Since in dimensions 5 and 6 we have an explicit list we get the following corollary (see [CGG16]).

Corollary 2.31. For every integer k > 4, every connected sum of the form

#(k − 3)(S2 × S3), (2.4.1)

(S2×˜ S3)#(k − 4)(S2 × S3), (2.4.2)

#(k − 4)(S2 × S4)#(k − 3)(S3 × S3), (2.4.3)

(S2×˜ S4)#(k − 5)(S2 × S4)#(k − 3)(S3 × S3), (2.4.4) 2.4 isometric actions 31 has a metric with positive Ricci curvature invariant under a cohomogeneity-two torus action. Remark 2.32. In dimension 4, this result was known by the work of Baza˘ıkinand Matvienko in [BkM07].

3 RIEMANNIANFOLIATIONS

In this chapter we review concepts and results of singular Riemannian foliations and discuss their relation to group actions. Our main references are [ABT13], [GGR15], [MM03], [Mol88], and [MR18].

3.1 singular riemannian foliations.

A Singular Riemannian Foliation on a Riemannian manifold M, which we denote by

(M, F), is the decomposition of M into a collection F = {Lp | p ∈ M} of disjoint connected, complete, immersed submanifolds Lp, called leaves, which may not be of the same dimension, such that (see [ABT13]):

(i) Every geodesic meeting one leaf perpendicularly, stays perpendicular to all the leaves it meets.

(ii) For each point p ∈ M there exist local smooth vector ﬁelds spanning the tangent space of the leaves.

If (M, F) satisfies the first condition, then we say (M, F) is a transnormal system. If it satisfies the second one, we say (M, F) is a singular foliation. When the dimension of the leaves is constant, we say the foliation is a regular Riemannian foliation or just a Riemannian foliation. In the remarks at the end of [Wil07] it is stated that for a Riemannian manifold M, if F is a partition which is a transnormal system,

33 34 riemannian foliations

then there are Lipschitz continuous vector ﬁelds spanning the tangent spaces of the leaves, i.e. F is a Lipschitz foliation. It is a question of interest to know if this can be improved in the following sense: Question. Is any transnormal system a smooth foliation, that is a foliation where the vector ﬁelds spanning it are smooth?

These are some examples of singular Riemannian foliations:

1. Given (M, g) a Riemannian manifold, deﬁne a singular foliation by letting each point be a leaf. This will be a singular Riemannian foliation, which is one of the two trivial foliations. The other one is taking the foliation given by one single leaf, namely M.

2. If (M, F) is a regular Riemannian foliation, then the foliation we obtain by taking the closure of the leaves, denoted by F, is a singular Riemannian foliation (see [ABT13, Mol88]).

3. If G is a compact Lie group acting on M by isometries, then the orbits of the action give a singular Riemannian foliation. These foliations are called homogeneous foliations (see section 3.4).

Another family of interesting examples is given by isoparametric submanifolds. Given an immersed isoparametric submanifold L ⊂ M, i.e. a codimension one submanifold with constant mean curvature, we can partition the ambient manifold into the submanifolds parallel to L, which are all isoparametric unless they lie on the focal set of L. In this case they have lower dimension. This partition gives a singular Riemannian foliation of M (see [AR16])

We say that a singular Riemannian foliation is closed if all the leaves are closed. The dimension of a foliation F, denoted by dim F, is the maximal dimension of the 3.1 singular riemannian foliations. 35 leaves. The codimension of a foliation is the codimension of the maximal dimensional leaves, that is, codim(M, F) = dim M − dim F.

The leaves of maximal dimension are called regular leaves and the leaves that do not have maximal dimension are called singular leaves. Points on regular leaves are called regular points and points on singular leaves are called singular points. Since F gives a partition of M, for each point p ∈ M there is a unique leaf, which we denote by Lp, that contains p and we say Lp is the leaf through p. Furthermore as with group actions, from the partition, we can consider the quotient space M/F which we call the leaf space. The quotient map π : M → M/F associated to it, is the leaf projection map. As with group actions we will denote the image of a subset N of M under the projection map π by N ∗. For a point p ∈ M, observe that ∗ ∗ p = Lp, by the definition of π. Since M carries by definition a Riemannian metric g, in the case where the singular Riemannian foliation F is closed, the quotient map π induces a metric d∗ on M ∗. With this metric the quotient π becomes a submetry. Recall that a submetry is a map that for any point p and ε > 0 small enough, it sends Bε(p), the ball of radius ε > 0 around p, to Bε(π(p)). Furthermore, for a closed singular Riemannian foliation (M, F), if the Riemannian manifold (M, g) is complete and has sectional curvature bounded below by λ ∈ R, then (M ∗, d∗) is an Alexandrov space of curvature also bounded below by λ (see for example [LT10, BBI01]). This means there M ∗ is a complete length space, and the curvature is defined via comparison triangles.

For a singular Riemannian foliation (M, F), the foliation gives a stratiﬁcation of

M. For k 6 dim F we deﬁne the k-dimensional stratum as:

Σ(k) = {p ∈ M | dim Lp = k}. 36 riemannian foliations

The regular stratum Σreg = Σ(dim F) is an open, dense and connected submanifold of M (see [Rad12, Lemma 2.2.2]). The foliation restricted to the regular stratum ∗ yields a Riemannian foliation (Σreg, F), and Σreg is open and dense in the leaf space M ∗. Furthermore by Proposition 3.7 in [Mol88], if (M, F) is a singular Riemannian ∗ foliation with compact closed regular leaves, then Σreg is an orbifold. Note that

the foliation is regular if and only if Σreg = M. A leaf L ⊂ M is called regular if dim L = dim F, and singular otherwise.

To close this section we mention an interesting type of singular Riemannian foliations, called polar foliations (some authors refer to them as singular Riemannian foliations with sections), which has recently attracted the attention of some authors (see for example [Ale04, AG07, ABT13]). A singular Riemannian foliation (M, F)

is polar if, for each regular point p in M, there is an immersed submanifold Σp containing p, called a section through p, such that its dimension is equal to the codimension of the foliation, it intersects all the leaves, and it is orthogonal to all the leaves. When the polar foliation is given by a group action (i.e. it is homogeneous), we say it is a polar group action. When we consider the distribution D normal to the leaves on M, then by the following theorem the condition of being polar is equivalent to D being an integrable distribution. Theorem 3.1 (Theorem 1.4 in [Ale06]). Let F be a singular Riemannian foliation on a complete Riemannian manifold M. If the normal distribution D is integrable, then F is polar and the set of regular points is open and dense in each section.

3.2 infinitesimal foliation.

An important tool in the study of singular Riemannian foliations is the inﬁnitesimal foliation. Let M be a complete Riemannian manifold with a closed singular Rie- 3.2 infinitesimal foliation. 37 mannian foliation F. Given a point p ∈ M we will construct a singular Riemannian foliation on νpLp, the normal space of the leaf through p. We start by ﬁxing ε > 0, and considering Sp = expp(νpLp) ∩ Bε(p), where Bε(p) ⊂ M is the ball of radius ε centered at p. The foliation F induces a foliation F|Sp on Sp by setting the leaves of F|Sp to be the connected component of the intersection between the leaves of

F and Sp. This foliation may not be a singular Riemannian foliation with respect to the induced metric of M on Sp, i.e the leaves of F|Sp may not be equidistant with respect to the induced metric. Nevertheless from, [Mol88, Proposition 6.5], the ∗ pull-back foliation expp(F|Sp ) is a singular Riemannian foliation on νpLp ∩ Bε(0) equipped with the euclidean metric. Theorem 3.2. Let (M, F) be a singular Riemannian foliation, on a compact manifold, and ﬁx p ∈ M such that the leaf passing through p is just the point p. Then

(Sp, F|Sp ) is a singular Riemannian foliation, with the ﬂat metric on νpLp pulled back to Sp via the exponential map expp.

ε In an equivalent way, writing νpLp = νpLp ∩ Bε(0) and considering the pull- p ∗ ε p back foliation F = expp(F|Sp ), the space (νpLp, F ) is a singular Riemannian foliation with respect to the Euclidean metric of νpLp, which we denote as the infinitesimal foliation at p. With this description we define for small λ a homothetic ε λε ε λε transformation of hλ : νp → νp , by simply sending a vector in νp to νp . By the ε p following lemma, the foliation (νpLp, F ) is invariant under homotheties that fix the origin.

Lemma 3.3 (Lemma 6.2 in [Mol88]). The homothetic transformations hλ preserves the foliation F p.

We note that the origin {0} ⊂ νpLp is always a leaf of the infinitesimal foliation F p. Since by definition the leaves of F p stay at a constant distance from each other, the fact that the origin is a leaf implies that any leaf of F p is at a constant distance from the origin, and thus it is contained in a sphere around the origin. From this last fact it follows that we may consider the infinitesimal foliation restricted to the unit 38 riemannian foliations

⊥ ⊥ p normal sphere, which we denote by Sp , yielding a foliated round sphere (Sp , F ) ⊥ with respect to the standard round metric of Sp which is also called the infinitesimal ⊥ p foliation. From here on when we say “infinitesimal foliation” we refer to (Sp , F ). p ⊥ p We note that we may refer to (νpLp, F ) and (Sp , F ) as the infinitesimal foliation p indistinctly since (νpLp, F ) is invariant under homothetic transformations and thus ⊥ p it can be recovered from (Sp , F ).

A singular Riemannian foliation (M, F) such that for any point p ∈ M the ⊥ p infinitesimal foliation (Sp , F ) is polar is called an infinitesimally polar foliation. By Theorem 4.10 (d) in [ABT13] it follows that polar foliations are infinitesimally polar foliations. Infinitesimally polar foliations have a leaf space with more regularity. Theorem 3.4 (Proposition 6.7 in [ABT13]). Let F be a closed singular Rieman- nian foliation on a complete Riemannian manifold M. The leaf space M/F is a Riemannian orbifold if and only if F is infinitesimally polar.

Infinitesimally polar foliations characterize singular Riemannian foliations which can be covered by a regular Riemannian foliation, or, n other words, by singular foliations where we can resolve the singularities, without losing the transverse geometry. Formally we say that a regular Riemannian foliation (Mˆ , Fˆ ) is a geometric resolution of a singular Riemannian foliation (M, F), if there is a smooth surjective map F : Mˆ → M mapping leaves of Fˆ to leaves of F, and preserving the transverse lengths (see [Lyt10]). By the following theorem, infinitesimally polar foliations are the only foliations which admit a geometric resolution. Theorem 3.5 (Theorem 1.1 in [Lyt10]). A singular Riemannian foliation (M, F) has a geometric resolution if and only if F is infinitesimally polar. Furthermore, when F is infinitesimally polar there is a canonical resolution. Remark 3.6. We will see in the next section that for a singular Riemannian foliation

(M, F), we can compare two leaves Lq and Lp, which are close, via a ﬁbration whose connected components are the leaves of the inﬁnitesimal foliation. 3.3 holonomy and types of leaves 39

3.3 holonomy and types of leaves

In this section we define the holonomy group ΓL of a closed leaf L of a singular Riemannian foliation (M, F). The interpretation of the holonomy is the following: In Section 3.2 we defined the infinitesimal foliation which gives a description of the foliation around a point p in the fixed leaf L. But since the infinitesimal foliation is given by taking connected components of intersections of leaves of F with a ball around p, it might happen that two of these connected components are contained in a common leaf of F. This means that to recover a small neighborhood of p∗ ∈ ⊥ p M/F, we can consider first the quotient space Sp /F , and then identify the leaves ∗ ⊥ p L ∈ Sp /F which where contained in a leaf of F. This identification is done ⊥ p via an action of π1(L, p) on Sp /F , which we call the holonomy action. All of this discussion is encoded in the following theorem from [MR18]. It is the analog to Theorem 2.11 for singular Riemannian foliations, describing a tubular neighborhood of a leaf in the foliation. Theorem 3.7 (Slice theorem for singular Riemannian foliations, Theorem A in [MR18]). Let (M, F) be a singular Riemannian foliation, and let L be a closed ⊥ p leaf with infinitesimal foliation (Dp , F ) at a point p ∈ L. Then there is a group ⊥ K of foliated isometries of (Dp , Fp) and a principal K-bundle P over L, such that for small enough ε > 0, the ε-tube U around L is foliated diffeomorphic to ⊥ p (P ×K D , P ×K F ).

Let (M, F) be a singular Riemannian foliation and let p be a point contained in ⊥ p L, a closed leaf. Recall from Section 3.2 that the inﬁnitesimal foliation (Sp , F ) is invariant under homotheties (see Lemma 3.3). Using this property we can extend the inﬁnitesimal foliation to the whole normal space νpL ⊂ TpM of L at p. Given a path γ : [0, 1] → L starting at p, the following theorem gives us a foliated transformation from νpL to the normal bundle νL. 40 riemannian foliations

Theorem 3.8 (Corollary 1.5 in [MR18]). Let L be a closed leaf of a singular Rieman- nian foliation (M, F) , and γ : [0, 1] → L a piecewise smooth curve with γ(0) = p.

Then there is a map G: [0, 1] × νpL → νL such that:

(i) G(t, v) ∈ νγ(t)L for every (t, v) ∈ [0, 1] × νpL.

(ii) For every t ∈ [0, 1], the restriction G: {t} × νpL → νγ(t)L is a linear

isometry preserving the leaves of νL.

(iii) For every s ∈ R the map expγ(t)(sG(t, v)) belongs to the same leaf as

expp(sv).

Proof. See Corollary 1.5 in [MR18], or AppendixA.

Thus if we have a loop γ at p, from Theorem 3.8 we have a foliated linear isometry ⊥ p G: {1} × νpL → νpL, which we will denote by Gγ. We denote by O(Sp , F ) the group of foliated isometries of the infinitesimal foliation, i.e. all the isometries which preserve the foliation. We note that such an isometry may map a leaf to a different leaf. By O(F p) we denote the foliated isometries which leave the foliation invariant, ⊥ p ⊥ p i.e. the isometries f ∈ O(Sp , F ) such that for any leaf L of (Sp , F )s we have ⊥ p ⊥ p f(L) ⊂ L. There is a natural action of O(Sp , F ) on the quotient Sp /F . The p kernel of this action is O(F ). In AppendixA we will show that if two loops, γ1 −1 ⊥ p and γ2, are homotopic then Gγ1 ◦ Gγ2 are in the kernel of the action of O(Sp , F ) ⊥ p ⊥ p on Sp /F . Therefore we obtain a group morphism from π1(L, p) to O(Sp , F ). Proposition 3.9. Let (M, F) be a singular Riemannian foliation, L a closed leaf of the foliation and p ∈ L. There is a well defined group morphism,

⊥ p p ρ: π1(L, p) → O(Sp , F )/O(F ),

given by ρ[γ] = [Gγ].

Proof. See Corollary A.5 in AppendixA. 3.3 holonomy and types of leaves 41

For a closed leaf L of a singular Riemannian foliation (M, F) we deﬁne the ⊥ p p holonomy of the leaf L as the image ΓL < O(Sp , F )/O(F ) of π1(L, p) under the morphism ρ. When we consider the holonomy of a leaf Lp trough a point p ∈ M, we will denote it by Γp. We say that a regular leaf L is called a principal leaf if the holonomy is trivial, and exceptional otherwise.

The holonomy action can be interpreted in the case of regular leaves, as in the work of Molino in [Mol88, Section 1.7], as follows. Suppose that (M, F) is a Rie- mannian foliation of dimension m on an n-manifold (for the general case consider

(Σreg, F)). Let L be a regular closed leaf and take γ : [0, 1] → L a path in L, ⊥ with p ∈ L as start point. When we consider for v ∈ Dp the germ of the map expγ(1)(Gγ(v)), we obtain an element of the holonomy as defined in [Mol88, Sec- tion 1.7], or [MM03, Chapter 2]. Furthermore we will see that for any path γ, the holonomy transformation associated to γ in [Mol88, MM03] is given by the germ of expγ(1)(Gγ(v)). In order to do this recall that for a Riemannian foliation (M, F), there exist a neighborhood U of p in M and a diffeomorphism φ: U → Rn, such that the image of the foliation (U, F) is given by the preimages of the projection map Rn → Rn−m (see [Mol88, MM03]). For the sake of clarity we assume first that γ([0, 1]) ⊂ U. Since (U, F) is foliated diffeomorphic to (Rn−m × Rm, Rm), then we can extend the vector field γ0 to a vector field X on U, in such a way that it is invariant under rescalings. This implies that X is a linearized vector field (see AppendixA for the definition of linearized vector field). The holonomy element of γ is the germ of the map given by the flow of X (see Figure 3.1). As one can see in

AppendixA, this is exactly how the map Gγ is deﬁned. In the general case when there is a foliated atlas of (M, F), we can do the previous analysis on each chart. Thus this shows that the notion of holonomy for a singular Riemannian foliation is an extension of the notion of holonomy for Riemannian foliations. For more details on the theory of Riemannian foliations we invite the reader to consult [Mol88], and [MM03]. 42 riemannian foliations

Proposition 3.10 (Theorem 2.6 in [MM03]). For a Riemannian foliation (M, F), if L is a compact leaf, then its holonomy group is ﬁnite.

Figure 3.1.: Construction of holonomy action for a regular leaf.

From Proposition 3.10 we have that the holonomy of a compact regular leaf is ﬁnite. The following theorem shows that for a singular Riemannian foliation (M, F)

such that the Riemannian foliation (Σreg, F) has compact closed leaves, then the

leaf space Σreg/F is an orbifold. Theorem 3.11 (Theorem 2.15 in [MM03]). Let (M, F) be a Riemannian foliation such that any leaf of F is closed compact. Then the space of leaves M/F has a canonical orbifold structure of dimension q. The isotropy group of a leaf in M/F is its holonomy group.

If L is a principal leaf of a singular Riemannian foliation (M, F) on an n-dimensional manifold with all regular leaves compact closed, then Theorem 3.11 and the ∗ ∗ fact that Γp is trivial show that L is a regular point of the orbifold Σreg. With this we can easily see that when we consider the stratum of principal leaves, which we denote ∗ ∗ by Mprin, then Mprin corresponds to the manifold part of the orbifold Σreg. Thus

Mprin is open and dense in Σreg. Since in general for a singular Riemannian foliation ∗ ∗ Σreg is open and dense in M, then Mprin is open and dense in M . Furthermore, from the fact that the manifold part of an orbifold is connected (see for example 3.3 holonomy and types of leaves 43

∗ [Lan18, Lemma 2.3], [Yer14], [Fae11]) it follows that the set Mprin is connected in M ∗. Since it is locally euclidean, it is path connected.

We collect these observations in the following theorem. Theorem 3.12. For a singular Riemannian foliation (M, F) with compact closed ∗ regular leaves, the principal stratum Mprin is open dense in M, and Mprin is connected and path connected in the leaf space M ∗.

As stated at the end of Section 3.2, for a singular Riemannian foliation (M, F) ⊥ p the infinitesimal foliation (Sp , F ) at p ∈ M, gives rise to a way of comparing leaves of different types, which will be exploited in the present work. Given a fixed ⊥ point p ∈ M and a vector v ∈ Sp , set q = expp(εv). If ε is small enough, then

Lq is contained a tubular neighborhood of Lp, and thus there is a well defined smooth closest-point projection proj: Lq → Lp which is by Lemma 6.1 in [Mol88] is a submersion. The connected component of the fiber of proj through q can be p identified with the leaf Lv ∈ F , through v. Taking Lp = Lep/ proj∗(π1(Lq)), the quotient of the universal cover Lep of Lp, we have a finite cover Lp → Lp such that proj: Lq → Lp lifts to a fibration:

ξ Lv → Lq → Lp. (3.3.1)

Clearly fibration (3.3.1) is a surjective map by construction. The following proposition gives another way of obtaining the covering L of L, via a subgroup H of the holonomy group Γp. ⊥ ∗ Proposition 3.13. For v ∈ Sp with image v ∈ Sp∗ , set H to be the subgroup of ∗ Γp fixing v . Then, taking q = expp(v), the finite cover Lp of Lp in the fibration

ξ : Lq → Lp is Lep/H. 44 riemannian foliations

−1 Proof. Let F = proj (p) be the ﬁber of the metric projection proj: Lq → Lp, which may consist of several connected components. The long exact sequence of the ﬁbration looks like

proj∗ ∂ · · · → π1(F , q) → π1(Lq, q) −→ π1(Lp, p) → π0(F , q) → 0.

From exactness, we conclude that (proj∗)(π1(Lq, q)) = ker(∂). We recall how the

map ∂ : π1(Lq, q) → π0(F , q) is defined, following a modification of the definitions 1 1 presented in Hatcher (see Sections 4.1 and 4.2 in [Hat10]). Let λ1 : D → S be 1 0 1 the map collapsing ∂D to a point. Let δ0 : S → D be the inclusion as the 1 boundary. Consider a loop ϕ: S → Lp with base point p. By the homotopy 1 1 lifting property there is a lift λ: D → Lq for the map ϕ ◦ λ1 : D → Lp, with

λ(0) = q. Furthermore, by deﬁnition, we have that ϕ ◦ λ1 ◦ δ0 = proj ◦λ ◦ δ0 is

constant. Therefore the image of the map ψ = λ ◦ δ0 is contained in F . Thus we 0 1 have a map ψ : S → F . We deﬁne ∂[ϕ] = [ψ]. Let α: S → Lp be a loop in

proj∗(π1(Lq, q)) = ker ∂. Consider Gt = G: {t} × νpLp → νLp, the transformation

given by Theorem 3.8 corresponding to α. Then α˜ (t) = expα(t)(Gt(v)) is a lift of α

in Lq. Since ∂ does not depend on the choice of a lift we have that 0 = ∂[α] = [α˜ ]. It follows that the end point of α˜ : [0, 1] → F is in the same connected component

of F as q. Therefore we have that G1 : νpLp → νpLp ﬁxes the inﬁnitesimal leaf Lv ⊥ in Sp /Fp. Thus proj∗(π1(Lq, q)) ⊂ H. Conversely, if we start with [α] ∈ H, then

for the map G: [0, 1] × νpLp → νLp given by Theorem 3.8, we have G1 maps the

inﬁnitesimal leaf Lv to itself. By deﬁnition this means that expp(G1(v)) is in the

same connected component of F as q = expp(v). Theorem 3.8(iii), implies that the

path expp(Gt(v)) is a path between q and expp(G1(v)) in F . Thus we have that

∂[α] = 0. Therefore we conclude that proj∗(π1(Lq, q)) = H.

In particular Proposition 3.13 gives a way to detect if there is holonomy for a closed leaf of (M, F). 3.4 homogeneous foliations 45

If the map ξ : Lq → Lp is proper then the following theorem of Ehresmann says that ξ is a locally trivial fibration. Theorem 3.14 (Ehresmann’s fibration lemma, in [Ehr51]). If W and N are smooth manifolds, and f : W → N is a smooth surjective submersion which is also proper, then f is a locally trivial fibration. This means that for each point p ∈ N there exists an open neighborhood U ⊂ N of p, and a diffeomorphism φ: f −1(U) → U × F , where F = f −1(p), such that the following diagram commutes:

φ f −1(U) U × F

f πU U

As a particular case, if in the previous theorem W is a compact manifold, we have the following proposition: Corollary 3.15. Let W and N be smooth manifolds, with W compact. If f : W → N is a smooth surjective submersion, then f is a locally trivial fibration. Remark 3.16. We note that in Ehresmann’s lemma the fiber bundle given by the projection map f, may not have as structure group a Lie group, but rather a very large topological group, namely the diffeomorphism group of the fiber, Diff(F ). Although Diff(F ) is in general not a Lie group, it is a Frobenious group, i.e. the group operations are smooth with respect to a Frobenious atlas (see [GW07]).

3.4 homogeneous foliations

A classical set of examples of singular Riemannian foliations comes from smooth actions of compact Lie groups on smooth manifolds. Let G be a compact Lie group acting smoothly on a smooth manifold M, and let us assume it acts eﬀectively. Given 46 riemannian foliations

any Riemannian metric g on M, we can always construct a Riemannian metric gG

which is invariant under the action of G, i.e. G < Isom(M, gG) (see [AB15, Proof of Slice Theorem 3.49]). The Lie algebra g of the group G provides the vector ﬁelds spanning the tangent spaces of the leaves, which in this case are the orbits of the action, so the partition induced by the orbits of G is a foliation. Furthermore, by the following proposition the foliation (M, G) is a transnormal system, and thus a singular Riemannian foliation. Proposition 3.17 (Proposition 3.78-(i) in [AB15]). If γ is a geodesic which starts normal to the orbit G(γ(0)), then γ(t) is normal to the orbit G(γ(t)) for all times.

Proof. Since the action of G is by isometries, any vector field X which is tangent to orbits is a Killing vector field. So it is sufficient to show that for a geodesic γ : I → M, if a Killing vector field X is orthogonal to γ0(0), then X is orthogonal 0 to γ (t) for all t. Since a vector field X is Killing if and only if g(∇Y X, Z) = 0 −g(∇ZX, Y ) for all vector fields Y and Z, then in particular g(∇γ0(t)X, γ (t)) = 0 d 0 0 and so dt g(X, γ (t)) = 0. Thus g(X, γ (t)) is constant, and since at t = 0 it is zero, we conclude that X and γ0(t) are orthogonal.

A singular Riemannian foliation (M; F) is called homogeneous if it is induced by a group G acting by isometries in M. By the work of Radeschi in [Rad14], there are several examples of non-homogeneous singular Riemannian foliations. In general, given a singular Riemannian foliation it is a diﬃcult problem to show it is either homogeneous or non-homogeneous.

⊥ p In the case of homogeneous foliations the inﬁnitesimal foliation (Sp , F ) on the normal sphere at p is given by connected components of the orbits of the action of ⊥ 0 Gp in Sp via the isotropy representation. Therefore, denoting by Gp the connected

component of Gp containing the identity element, the inﬁnitesimal foliation is given 0 ⊥ by considering only the action of Gp on Sp given by the isotropy representation. 0 The holonomy of G(p) is given by Gp/Gp (see [MR18, Section 3.1]). For q ∈ M 3.4 homogeneous foliations 47

close to p with Gq a subgroup of Gp, the ﬁber bundles given by (3.3.1) are of the form: 0 0 Gp/Gq → G/Gq → G/Gp,

0 where Gp is the connected component of the identity of the isotropy group Gp, and 0 G/Gp is a cover of the orbit G/Gp (see [GGR15, Example 2.4]).

Part II

ASPHERICALFOLIATIONS

4 CROSS-SECTIONSAND A-FOLIATIONS

In this chapter we will consider A-foliations, i.e. singular Riemannian foliation with aspherical leaves. We will focus on the case of A-foliations of codimension 2 on an (n + 2)-dimensional, compact, simply-connected manifold M. It has been proven in [GGR15, Corollary B] that the leaves of such foliation must be homeomorphic to tori. We exploit this fact to show that several results of torus actions extend to A-foliations. We begin by giving conditions to be able to compare in a foliated sense two foliated manifolds via their leaf spaces.

4.1 cross-section for the leaf space

For homogeneous foliations given by a ﬁxed group G, the existence of cross-sections for the projection map π : M → M/F has been exploited in works such as [Oh83a], [OR70], [OR74], to classify up to homeomorphism the manifolds M admitting a G-action, via the orbit space M ∗. In order to get the classiﬁcation up to equivariant homeomorphism, more structure is needed on M ∗ which will be discussed in following sections.

In this section we state suﬃcient topological conditions on the leaf space of a foliated manifold (M, F) for the existence of a cross-section of the projection map π : M → M/∗.

51 52 cross-sections and a-foliations

We start by recalling the relative lifting problem. Given a relative CW-complex (W , A), continuous maps p: X → Y , g : W → Y and f : A → X, we must ﬁnd suﬃcient conditions to guarantee the existence of an extension f˜: W → Y of f lifting g, i.e. a map f˜ which makes the following diagram commute:

f A X f˜ i p (4.1.1) g W Y

This problem is well understood when the map p is a ﬁbration (e.g. [Hat10, DK01]). In a more general setting, when p is not a ﬁbration, one can ask that the spaces be nicely behaved to get a family of obstructions.

We recall that a topological space F is called n-simple if it is path connected,

with Abelian fundamental group π1(F ), and the action of the fundamental group

on all the higher homotopy groups πk(F ) is trivial. This last condition is equivalent k to πk(F ) = [S , F ]. A space which is simple for all n is called simple.

For a pair of spaces (W , A), with A path connected, we can deﬁne the rela- k k−1 tive homotopy groups πk(W , A) by considering I = I × I, and considering Ik−1 as the face Ik−1 × {0}. Set Jk−1 to be the closure of ∂Ik \ Ik−1, and for

a base point x0 ∈ A, deﬁne πk(W , A, x0) as the set of homotopy classes of maps k k k−1 (I , ∂I , J ) → (W , A, x0). Since W and A are path connected spaces, the last deﬁnition does not depend on the choice of the base point, just as in the non-relative

case. Furthermore, there is a natural action of π1(A) on πk(W , A). Consider a loop k k k−1 γ : I → A with γ(0) = x0, and a map f : (I , ∂I , J ) → (W , A, x0). Consider the map γf deﬁned as in Figure 4.1, where for a point on a radial line γf is just γ

(see [Hat10, Section 4.1]). If this action of π1(A) on πn(W , A) is trivial, we say that (W , A) is an n-simple pair. A pair that is n-simple for all n is called a simple pair (see [Hat10, Section 4.1][DK01, Section 6.16]). 4.1 cross-section for the leaf space 53

γ A γ

Figure 4.1.: Map γf

We recall some basic topological constructions which will be used frequently in the remainder of the section. We denote by Y I the space of all continuous paths γ : I → Y with the compact-open topology. There is a natural fibration q : Y I → Y called the path space fibration, with q defined as q(γ) = γ(0). For a map p: X → Y , the mapping path fibration πp : Eπp → Y is the fibration with total space the total space p∗(Y I ) ⊂ X × Y I of the pullback via p. For a path γ : I → Y , with γ(0) = p(w) the projection map πp is defined as πp(w, γ) = γ(1). The following theorem gives some properties of the construction πp : Eπp → Y . Theorem 4.1 (Theorem 6.18 in [DK01]). Suppose that p: X → Y is a continuous map.

1. There exists a homotopy equivalence h : X → Eπp so that the diagram,

h X Eπp

p πp Y

commutes.

2. The map πp : Eπp → Y is a ﬁbration.

3. If p: X → Y is a ﬁbration, then h is a ﬁber homotopy equivalence.

The fiber Fp of πp is called the homotopy fiber of p, and there is a homotopy equivalence between X and Eπp . If p is already a fibration with fibers F , then Fp is homotopy equivalent to F . Next we define the mapping cylinder Mp. This is the 54 cross-sections and a-foliations

space obtained by considering the disjoint union of X × I with Y , and identifying

(x, 1) with p(x). There is an inclusion i: X → Mp given by i(x) = [x, 0]. The properties of the mapping cylinder are given by the following theorem. Theorem 4.2 (Theorem 6.27 in [DK01]). Let p: X → Y be a continuous map, and

let i: X → Mp be the inclusion deﬁned above.

1. There exists a homotopy equivalence h: Mp → Y such that the following diagram commutes: X p i

Y M h p

2. The inclusion i: X → Mp is a coﬁbration.

Recall that a map i: X → Y is a coﬁbration if the following diagram has a solution for any space Z:

X × {0} X × I

i Z i×Id

Y × {0} Y × I

Given a map f : X → Y between path-connected spaces, a Moore-Postnikov tower for f is a collection of spaces,

αn · · · → Zn+1 → Zn → · · · → Z1,

and continuous maps αn : Zn+1 → Zn, λn : X → Zn, µn : Zn → Y such that:

(i) αn ◦ λn+1 = λn;

(ii) µn ◦ αn = µn+1;

(iii) for i < n the map λn induces an isomorphism between πi(X) and πi(Zn) and a surjection for i = n; 4.1 cross-section for the leaf space 55

(iv) for i > n the map µn induces an isomorphism between πi(Zn) and πi(Y ) and an injection for i = n;

(v) the map αn is a ﬁbration with ﬁber an Eilenberg-MacLane space

K(πn(F ), n), where F is the homotopy ﬁber of f.

The points (i) to (v) can be summarized in the following commutative diagram:

. .

α3

λ3 µ3 Z2

X Z1 Y

The idea behind a Moore-Postnikov tower is to have a series of ﬁbrations with spaces which start approximating the homotopy type of Y and gradually they ap- proximate the homotopy type of X. In general a ﬁbration F → E → B is called principal if there is a commutative diagram of the form:

F E B

ΩB0 F 0 E0 B0

Here the second row is a fibration sequence, and all vertical maps are weak homotopy equivalences, i.e. they induced isomorphisms between homotopy groups of all degrees. When the fibrations αn : Zn+1 → Zn are principal fibrations we say that we have a Moore-Postnikov tower of principal fibrations. In general any map between CW-spaces admits a Moore-Postnikov tower, but the following theorem explicitly states when does a Moore-Postnikov tower of principal fibrations exist. 56 cross-sections and a-foliations

Theorem 4.3 (Theorem 4.71 in [Hat10], Existence of Moore-Postnikov tower of principal ﬁbrations). For a given map f : X → Y between connected CW-spaces, a

Moore-Postnikov tower of principal ﬁbrations exists if and only if π1(X) acts trivially

on πn(Mf , X) for all n > 1, where Mf is the mapping cylinder of f.

With all these concepts at hand we can now give the general solution to the relative lifting problem. The presence of a Moore-Postnikov towers is useful for solving the relative lifting problem associated to Diagram (4.1.1).

The idea is to inductively construct a lift W → Zn for each n, and from these lifts obtain a lift W → X. We start with the case where the map p: X → Y is a fibration. Theorem 4.4 (Obstruction Theory in [Hat10]). Let p: X → Y be a fibration with fiber F , (W , A) a CW-pair with W simply connected. Assume the fibration has a Moore-Postnikov tower of principal fibrations and consider the relative lifting problem:

f A X f˜ i p g W Y

n+1 There exists an obstruction ωn ∈ H (W , A; πn(F )), such that a lift f˜: W → X

extending f : A → X exists, if ωn = 0 for all n.

Proof. First we note that since we have a ﬁbration p: X → Y , we may take Z1 to be

the covering space of Y corresponding to the subgroup p∗(π1(X)) of π1(Y ). Since

W is simply-connected we can lift g to W → Z1, which agrees with g ◦ λ1 : A → Z1. Since the Moore-Postnikov tower is by principal ﬁbrations, for the inductive step we have a commutative diagram as follows:

A Zn PK

W Zn−1 K = K(πn(F ), n + 1). 4.1 cross-section for the leaf space 57

Here PK → K is the path fibration, defined by fixing a point b0 ∈ K, and letting

PK be the space of all curves in K starting at b0, and the letting PK → K be the map that sends each path to its end point. Since Zn is the pullback, the elements in Zn are pairs consisting of a point in Zn−1 and a path from its image in K to the base point in K. A lift W → Zn therefore amounts to a nullhomotopy of the composition W → Zn−1 → K. Since we have already deﬁned such a lift on A, we have a nullhomotopy of A → K, and the desired nullhomotopy of W → K must extend this nullhomotopy on A. The map W → K together with the nullhomotopy on A gives a map W ∪ C(A) → K, where C(A) is the cone of A. Since K is an

Eilenberg-MacLane space K(πn(F ), n + 1), the map W ∪ C(A) → K determines the desired obstruction

n+1 n+1 ωn ∈ H (W ∪ C(A); πn(F )) = H (W , A; πn(F )).

If ωn = 0, by construction we have that there is a nullhomotopy of W → K extending the given nullhomotopy A → K.

If we succeed in extending the lifts A → Zn to lifts W → Zn for all n, then we obtain a map W → lim Z , to the inverse limit lim Z , extending the given ←− n ←− n A → X → lim Z . Let M be the mapping cylinder of X → lim Z . From the ←− n ←− n hypothesis that the restriction of W → lim Z ⊂ M to A factors through X, this ←− n gives a homotopy of this restriction to the map A → X ⊂ M. We extend this homotopy to a homotopy of W → M producing a map (W , A) → (M, X). Since the map X → lim Z is a weak homotopy equivalence, then π M, X 0 for ←− n i( ) = all i, and from the so-called Compression Lemma (see Lemma 4.6 in [Hat10]), we conclude that the map (W , A) → (M, X) is homotopic relative to A to a map W → X. Hence the map W → X extends the given map A → X.

Since in the previous theorem we started with a ﬁbration p: X → Y , we state the relevant existence of obstructions but for a general continuous map. 58 cross-sections and a-foliations

Theorem 4.5. (Obstruction to extension) Let (W , A) be a relative CW -complex, with W simply-connected, and assume we have continuous maps p: X → Y , f : A →

X and g : W → Y . Furthermore, suppose that the homotopy ﬁber Fp of p is simple,

and that (Mp, X) is a simple pair. Then the following are true:

k+1 (i) There is a family of obstructions ωk ∈ H (W , A; πk(Fp)) such that there

exists a lift f˜ of f solving diagram (4.1.1) if ωk = 0 for all k.

(ii) If Fp is an Eilenberg-McLane space K(π, `), then there is a unique obstruc- `+1 tion ω` ∈ H (W , A; π), and the lift f˜ of f solving diagram (4.1.1) exists if

and only if ω` = 0.

Proof. We sketch here the proof. For further details we invite the interested reader to see, for example, [Hat10, Chapter 4] for a more detailed discussion. First we

show that the pair (Mp, X) is simple only when the pair (Mπp , Eπp ) is simple. To prove this we start by noting that there is a homotopy equivalence between X and

the total space Eπp of the path space ﬁbration for p: X → Y given by Theorem 4.1.

For the maps p: X → Y and πp : Eπp → Y , from Theorem 4.2 we have coﬁbrations

j : X → Mp and i: Eπp → Mπp . Furthermore Theorem 4.2 also yields a homotopy

equivalence between the mapping cylinder Mp and Mπp . Combining the diagrams obtained by applying Theorems 4.1 and 4.2 to the map p: X → Y , and the diagram

obtained by applying Theorem 4.2 to πp : Eπp → Y we obtain the following diagram, which commutes up to homotopy:

X Eπp

j i

Mp Mπp

Observe that the arrows going down are coﬁbrations. Then from the previous com-

mutative diagram and [Bro06, 7.4.2], the homotopy groups πk(Mp, X) are (equiv-

ariantly under the action of π1(X)) isomorphic to πk(Mπq , Eπp ) (with the action of 4.1 cross-section for the leaf space 59

π1(Eπp )). Thus (Mp, X) is simple if and only if (Mπp , Eπp ) is simple. Therefore, for the fibration πp : Eπp → Y , there exists a Moore-Postnikov tower by principal fibrations which yield the desired family of obstructions. Thus we may apply Theorem 4.4 to the fibration πp : Eπp → Y .

We also note that we may apply the last argument in the proof of Theorem 4.4, and use the fact that the restriction of W → lim Z ⊂ M to A factors through X, ←− n to construct the lift W → X which extends A → X.

Remark 4.6. The reason why in Theorem 4.5(i) we have an “if... then...” statement and on Theorem 4.5(ii) we have an “if and only if” statement lies in the fact that for the proofs of these theorems we use a Moore-Postnikov tower of principal ﬁbrations

· · · → Z2 → Z1 → Y for p. In the case of Theorem 4.5(i) the lifts may be not unique, and in some examples this may yield non trivial ωk even when an extension exists. An exception to this, is the case when Fp is an Eilenberg-McLane space (see [Hat10, Section 4.3]). Remark 4.7. The condition of W being simply connected is used to ensure a unique lift from W to Z1 in the Moore-Postnikov chain. Remark 4.8. When we consider a principal S1-bundle p: X → Y , the only obstruction to a cross-section of p is the Euler class of the bundle (see [Mor01]). This class 2 1 coincides with the obstruction ω2 ∈ H (Y ; π1(S )) given by part (ii) in Theorem 4.5.

Thus for maps p: X → Y with homotopy ﬁber Fp aspherical, the obstruction obtained in Theorem 4.5(ii) is a generalized Euler class.

Given a singular Riemannian foliation (M, F), we consider the subset Mprin of ∗ M, consisting of principal leafs. The projection map M → M restricted to Mprin yields a ﬁbration: ∗ L → Mprin → Mprin.

We apply Theorem 4.5 to get a family of obstructions, which we will call ﬁrst obstructions. 60 cross-sections and a-foliations

Theorem A. Let (M, F) be a closed singular Riemannian foliation with M simply- connected, quotient map π : M → M ∗, and principal leaf L, which is simple and ∗ connected. Furthermore assume Mprin is simply-connected and (Mπ, Mprin) is simple. 1 k+1 ∗ Then there is a family of obstructions ωk ∈ H (Mprin; πk(L)) such that a cross- ∗ 1 section σ : Mprin → Mprin exists if ωk = 0 for all k.

∗ Proof. By applying Theorem 4.5 with X = Mprin, W = Y = Mprin, and A = ∅ we get the result.

Even if a section exists on the principal part of the foliation, it may happen that it cannot be extended to the whole leaf space (as an example see [Fin76] or [Fin77]). To solve this new extension problem we need another family of obstructions which we call second obstructions. Theorem B. Let (M, F) be a closed singular Riemannian foliation with M simply connected, and consider the quotient map π : M → M ∗. Furthermore assume ∗ that the homotopy ﬁber Fπ is simple, and setting A = Mprin, assume we have al-

ready deﬁned a cross-section σ : A → Mprin. Then there is a family of obstructions 2 k+1 ∗ ∗ ωk ∈ H (M , A; πk(Fπ)) such that a cross-section σ˜ : M → M extending σ exists 2 if ωk = 0 for all k.

Proof. Since M is simply-connected, then M ∗ is also simply-connected. We apply Theorem 4.5 to obtain the desired result.

∗ ∗ In particular, when we cannot distinguish M from Mprin from a homotopical view point, we get the following corollary from TheoremB. Corollary C. Let (M, F) be a closed singular Riemannian foliation on a simply- ∗ connected manifold. Suppose that there is a section σ˜ : Mprin → Mprin, and the that ∗ ∗ hypothesis of TheoremB are satisﬁed. If Mprin has the same homotopy type of M , then the cross-section σ˜ can be extended to a section σ. 4.1 cross-section for the leaf space 61

Remark 4.9. Since the holonomy is only deﬁned for closed leaves (see Section 3.3), we ask that the foliation is closed in order to ensure the existence of a principal stratum in TheoremA, TheoremB, and CorollaryC.

The following are particular applications of CorollaryC in the setting of group actions.

We begin by considering M a closed, simply-connected, smooth (n + 2)-manifold, with an effective and smooth Tn-action. From [Bre72], it follows that, the orbit space M ∗ = M/Tn is a 2-disk with all the isotropy information contained in the boundary of M ∗. Thus via CorollaryC, we recover Theorem 2.20, which was proved first by Orlik and Raymond in [OR70] for 4-manifolds, and extended to arbitrary dimensions by Oh in [Oh83a]. Theorem 4.10 ([Oh83a],[KMP74], [OR70],Theorem 2.20). Let M be a closed simply-connected smooth (n + 2)-manifold, with an effective and smooth Tn-action. Then there exists a cross-section σ : M ∗ → M.

∗ Proof. First we point out that the ﬁbers of the ﬁbration Mprin → Mprin are the group Tn, a K(Zn, 1) Eilenberg-McLane space (since they are connected and aspherical). Furthermore since the principal orbits all lie in the interior of M ∗, which is contractible, we may apply TheoremA(ii) to show that the only obstruction vanishes. ∗ ∗ Thus we have a section from σ : Mprin → Mprin. Second, we point out that M and ∗ Mprin have the same homotopy type since they both are contractible. Thus we can extend the section σ to the whole orbit space M ∗ by virtue of corollaryC.

We see that the previous argument works in general when the group G is an Eilenberg-McLane space, and the orbit space is a k-disk, with all the isotropy contained in the boundary. In particular we obtain the following result, which gen- eralizes [ES17, Theorem 1.3]. The original result for smooth, eﬀective, cohomogeneity three torus actions on closed, simply-connected 6-manifolds was proven in [McG76]. 62 cross-sections and a-foliations

Theorem 4.11. Let T k act smoothly and eﬀectively on a smooth, closed n-dimensional manifold M, such that the orbit space M ∗ is an (n − k)-disk. Furthermore, suppose that all interior points of the orbit space correspond to principal orbits and that points on the boundary of M ∗ correspond to non-principal orbits. Then the orbit map π : M → M ∗ admits a cross-section. Remark 4.12. In Theorem 4.11, since eﬀective torus actions have trivial principal isotropy, we could have said that the boundary of M/Tk consist of all orbits with non-trivial isotropy.

Last we present a general example which shows, in the setting of homogeneous foliations, why we are interested in the existence of cross-sections. Namely we expose a simple case where the presence of cross-section allows to lift a homeomorphism between orbit spaces to foliated homeomorphisms between the foliated manifolds. Theorem 4.13. Suppose that M and N are compact manifolds, with a homogeneous foliation, by a proper, eﬀective action by a ﬁxed Lie group G. Furthermore, ∗ ∗ suppose there exist cross-sections s1 : M → M, s2 : N → N. If there exists a homeomorphism f ∗ : M ∗ → N ∗,

that preserves the isotropy type of the orbits, then M and N are foliated homeomorphic.

∗ ∗ Proof. Take p ∈ M and set q = s1[p] and q˜ = s2(f [p]). Since f preserves the

isotropy type we have that Gq = Gq˜. Because G(q) = G/Gq we then have that

p = gGq for some g ∈ G. We now deﬁne f : M → N by f(p) = gGq˜ ∈ G(q˜). The action of G is continuous, therefore f is an equivariant homeomorphism.

Remark 4.14. Theorem 2.21 is a corollary of Theorem 4.13. 4.2 a-foliations 63

4.2 a-foliations

Since in the proof of Proposition 4.13 the isotropy information was used to give auxiliary points in the leaves that helped construct the homeomorphism, and in general, for foliations we do not have a natural choice of something that plays the same role as the isotropy information, it is not clear that a result similar to Proposition 4.13 exists for an arbitrary Riemannian foliation.

By assuming more conditions on the topology of the leaves, namely on the homotopy type of the leaves, we can deﬁne extra information on the leaf space that will be analogous to the weights deﬁned for torus actions.

In this section we discuss a particular type of foliations, for which the construction of these weights is possible. These foliations have been already studied by Galaz-Garc´ıaand Radeschi in [GGR15]. In particular the authors give a complete description of the leaf space for the codimension 2 case in [GGR15].

An A-foliation is a foliation where all the leaves are closed, connected, and aspherical, i.e. for n > 1 the n-th homotopy group of the leaves is trivial. The following corollary in [GGR15] shows that the principal leaves of an A-foliation on a compact, simply-connected, Riemannian manifold are homeomorphic to tori. Theorem 4.15 (Corollary B in [GGR15]). Let (M, F) be an A-foliation on a compact Riemannian manifold M. If M is simply-connected, then the regular leaves are homeomorphic to tori.

We recall that for q close to p in M with respect to the metric of (M, F), if Lq is a principal leaf and Lp is any leaf in M, then there is a ﬁbration:

L → Lq → Lp,(3.3.1) 64 cross-sections and a-foliations

where Lp = L˜ p/H is a ﬁnite cover of Lp, and L is a leaf in the inﬁnitesimal foliation F p (see Section 3.2). Using this description we will describe the topology of the other leaves types in an A-foliation.

First we consider the case when the leaves of the infinitesimal foliation ⊥ p (Sp , F ) are connected. In this case the finite covering Lp is trivial, i.e. Lp = Lp. Thus following the proof of Theorem 3.7 in [GGR15], we prove the following result: Proposition 4.16. Let F , M and N be topological manifolds, with F connected, and let F → M → N be a fibration. If M is homeomorphic to a torus, then F and N are tori.

Proof. Since M is aspherical we have from Theorem 3.7 in [GGR15] that F and N are also aspherical. From the long exact sequence of the ﬁbration we get:

0 → π1(F ) → π1(M) → π1(N) → 0.

Since π1(M) is an Abelian, torsion-free, ﬁnitely-generated group, and π1(F ) is a

subgroup of π1(M), then π1(F ) is an Abelian, torsion free, finitely generated group. Thus by classification of finitely generated Abelian groups and the the Borel con-

jecture F is homeomorphic to a torus. Now assume that π1(N) has torsion. Then

for some k ∈ Z, the cyclic group Zk acts freely on the contractible manifold N˜ .

Therefore it follows that N˜ /Zk is an Eilenberg-MacLane space K(Zk, 1). This con-

tradicts the fact that K(Zk, 1) has infinite cohomological dimension. Thus π1(N) is an Abelian, torsion-free, finitely generated group. Again by the classification of finitely generated Abelian groups and the Borel conjecture, N is homeomorphic to a torus.

Corollary 4.17. In an A-foliation all leaves with trivial holonomy are homeomorphic to tori. 4.2 a-foliations 65

In the case when the leaf Lp has non-trivial holonomy, applying Proposition 4.16 to fibration (3.3.1) we have that the covering Lp is homeomorphic to a torus. Thus, applying the long exact sequence of homotopy groups to the fibration Lp → Lp with finite fiber F , we get,

0 → π1(Lp) → π1(Lp) → π0(F ) → 0.

Therefore π1(Lp) is a ﬁnite extension of π0(F ) by π1(Lp). Assume that π1(Lp) is n not torsion-free, and recall that since Lp is a torus, we have L˜ p = R . Then there n exists a ﬁnite cyclic subgroup Zk acting on the contractible manifold L˜ p = R . As in the proof of Proposition 4.16 this contradicts the fact that the Eilenberg-

MacLane space K(Zk, 1) has inﬁnite cohomological dimension. Since π1(Lp) is n Z and F is ﬁnite, we have that π1(Lp) = G is a crystallographic group (see

[FH83, Section 6],[AK57],[Zas48]). Thus π1(Lp) is a Bieberbach group, since it is a torsion free crystallographic group. By theorem 6.1 in [FH83], for n 6= 3, 4, Lp is homeomorphic to a Bieberbach manifold. In Theorem 0.7 in [KL09], it is proved that the Borel conjecture is true in dimension 3.

From the previous discussion it follows that we have proved the following proposition: Proposition 4.18. The leaves (of dim 6= 4) with non-trivial holonomy of an A- foliation are homeomorphic to Bieberbach manifolds. Remark 4.19. In [GGR15] the authors deﬁne a B-foliation as an A-foliation with all leaves homeomorphic to Bieberbach manifolds. Since a torus is a Bieberbach space, it follows from Propositions 4.17 and 4.18 that any A-foliation is a B-foliation. Because of this fact, we will not distinguish them in this work. Corollary 4.20. In an A-foliation all leaves (of dim 6= 4) are homeomorphic to Bieberbach manifolds. 66 cross-sections and a-foliations

Remark 4.21. The diﬀeomorphism type of the leaves of an A-foliation may not be unique. If the leaves have trivial holonomy, i.e. are homeomorphic to tori, then for

dimensions k > 5, there exist diﬀerent smooth structures {Uα1 , ϕα1 }, {Uα2 , ϕα2 } on k k k the k-torus T , such that τ1 = (T , Uα1 , ϕα1 ) is homeomorphic (as a topological k k k k manifold) to τ2 = (T , Uα2 , ϕα2 ), but τ1 is not diﬀeomorphic to τ2 (see for example [HS70]) .

As a concrete example of this exotic phenomena, we may consider Σk an exotic sphere and the standard torus:

Tk = S1 × · · · × S1 . | {z } k-times

The manifold Tk#Σk is homeomorphic to Tk but not diﬀeomorphic to Tk (see Remark pp.18 in [FJ90] and Theorem 3 in [FJO07]).

We end this section by stating for q ∈ M, with the leaf Lq through q singular, ⊥ which is the homeomorphism type of the leaves of the infinitesimal foliation (Sp , F). Since they are connected it follows from Theorem 4.15 and Proposition 4.16 that the infinitesimal foliation is an A-foliation by tori. We collect this fact in the following corollary. Corollary 4.22. The infinitesimal foliations of an A-foliation are A-foliations on round spheres whose leaves are all homeomorphic to tori.

4.3 molino bundle

Next we introduce the so called Molino bundle of a Riemannian foliation. Let (S, F) be a Riemannian foliation of codimension q (recall that this means that the dimension of the leaves is constant). We consider the subbundle N(S, F) of 4.3 molino bundle 67 the tangent bundle TS, consisting of vector ﬁelds on S which are orthogonal to the leaves of the foliation F. This is called the normal bundle of the Riemannian foliation (S, F). The normal bundle N(S, F) is the orthogonal complement to the smooth distribution (and thus a bundle over S) given by the tangent spaces of the leaves of the foliation F. We denote by Sb the bundle of orthonormal frames of N(S, F), and call it the transverse orthonormal frame bundle or the Molino bundle of the Riemannian foliation (S, F) (some authors denote Sb by OF (S, F), see for example [MM03]). Note that the projection map p: Sb → S has the structure of a principal O(q)-bundle. Therefore we can lift the foliation F of S to obtain a regular foliation Fb on Sb, called the lifted foliation on the orthonormal frame bundle of the Riemannian foliation.

We describe some properties of the foliated Molino bundle (Sb, Fb). The projection map p: (Sb, Fb) → (S, F) is a foliated map by construction of Fb. The foliation F is closed if and only if Fb is a closed foliation. The restriction of the map p to a leaf

Lb of Fb is a covering map of the leaf L = p(Lb). The group of deck transformations of the covering p: Lb → L is the holonomy group ΓL. Thus if L is a principal leaf of (S, F), the leaf of Fb corresponding to L is diﬀeomorphic to L. For details on the construction of this foliation Fb, and proofs of all the previous statements we invite the reader to check for instance Example 4.19 in [MM03].

We will use other properties of the Molino bundle, which are stated in the following theorem: Theorem 4.23 (Molino’s structure theorem, Theorem 4.26 in [MM03], Theorem 10.1 in [Ton97], [Mol82]). Let (S, F) be a Riemannian foliation of codimension q on a compact, connected, Riemannian manifold. The following hold:

(i) There exists a manifold Wc with an O(q)-action and a ﬁber bundle πb : Sb →

Wc such that πb is O(q)-equivariant.

(ii) The ﬁbers of πb are the closure of the leaves of Fb. 68 cross-sections and a-foliations

(iii) Let F denote the singular Riemannian foliation on S given by the closure of the leaves of F and consider W = S/F, the space of leaves. Then W =

Wc /O(q) and, for the quotient maps πb : Sb → W and π : S → W , the following diagram commutes, p Sb S

πb π Wc W β

Next we consider the principal universal bundle EO(q) → BO(q) associated to O(q). For a Riemannian foliation (S, F) on a compact manifold, we deﬁne the

Borel constructions of the spaces Sb and Wc given by Theorem 4.23, as the quotient

manifolds SbO = (Sb × EO(q))/O(q) and WcO = (Wc × EO(q))/O(q). The action of O(q) on both products, Sb × EO(q) and Wc × EO(q), is given by the diagonal

action. By functoriality we obtain a ﬁber bundle πbO : SbO → WcO. The ﬁbers of

this bundle are homeomorphic to the ﬁbers of the bundle πb : Sb → Wc . If (S, F)

is closed, then this induces a regular foliation FbO on SbO. The projection map

pO : (SbO, FbO) → (S, F) is a foliated map. Again the restriction of pO to Lb, a

leaf of Fb, is a covering map pO : Lb → L, where L is the corresponding leaf of F. Furthermore from Theorem 4.23(iii), we get the following commutative diagram:

πbO SbO WcO

pcO βb

S π W

Since by construction O(q) acts freely on Sb, we have a ﬁber bundle

pO : SbO → S with ﬁbers EO(q). Since EO(q) is contractible, pO is a homotopy

equivalence between SbO and S. Namely we have proved the following proposition. Proposition 4.24 (Proposition 2.4 in [FGLT15]). For a singular Riemannian foli-

ation (S, F) on a compact manifold, the Borel construction SbO of the Molino bundle Sb is homotopy equivalent to S. 4.3 molino bundle 69

Remark 4.25. By Theorem 3.11, if (S, F) is a closed Riemannian foliation on a compact manifold, the space of leaves S∗ = S/F is an orbifold. Furthermore, the foliation Fb on Sb is closed. Therefore the ﬁbers of πb : Sb → Wc are the leaves of

F. Since the fibers of πb : Sb → Wc are covering spaces of the fibers of F, they are diffeomorphic via the map p: Sb → S to the principal leaves of F. Moreover for a closed Riemannian foliation (S, F) on a compact manifold S, the Borel construction ∗ ∗ WcO coincides with Haefliger’s classifying space B(S ) of the orbifold S (see [ALR07, Corollary 5.2] [Hae84, Section 4], [FGLT15]). Therefore for n > 1 we have that orb ∗ πn (S ) = πn(WcO).

Consider L0 and L principal leaves in a closed singular Riemannian foliation ∗ (M, F) on a compact, simply-connected manifold M. Take any path γ : I → Σreg, ∗ ∗ with γ(0) = L0 and γ(1) = L . For a ﬁxed point x ∈ L0, from Proposition 1.3.1 x x in [GW09] there exists a unique curve γ : I → Σreg, such that γ (0) = x and it is perpendicular to all the leaves of F it intersects. Such a curve γx is called the horizontal lift of γ through x (see [GW09, Chapter 1] for more details). With this x we are able to deﬁne a homeomorphism hγ : L0 → L, by setting hγ(x) = γ (1). We ∗ ∗ will show that if we consider two such curves γ0 and γ1 connecting L0 to L , then the homeomorphisms hγ0 and hγ1 are homotopic.

In order to proof this, we need to deﬁne a full singular Riemannian foliation. We say that a singular Riemannian foliation F on a Riemannian manifold M is full, if for each leaf L there is some > 0, such that, the map v 7→ exp(v) is deﬁned for any unit vector v in the normal bundle νL of L. If M is complete this is the case. If F is a full singular Riemannian foliation on a Riemannian manifold M with all leaves closed, then M ∗ is a metric space, with a natural inner metric that has curvature locally bounded below in the sense of Alexandrov (see [LT10]). Finally, a full regular Riemannian foliation is simple, i.e. has closed leaves with trivial holonomy, if and only if the quotient M ∗ is a Riemannian manifold. (see [Lyt10]). In particular for full foliations on simply-connected manifolds we have the following result. 70 cross-sections and a-foliations

Lemma 4.26 (Corollary 5.3 in [Lyt10]). Let (M, F) be a full singular Riemannian foliation on a simply-connected Riemannian manifold M, with all the leaves closed.

Then the quotient B = Σreg/F, of the restriction of F to the regular part Σreg is a orb Riemannian orbifold with π1 (B) = 1.

∗ With the previous lemma we can then show that the quotient Mprin of the

principal stratum Mprin is simply connected. Lemma 4.27. Let (M, F) be a singular Riemannian foliation with closed leaves on ∗ a compact, simply-connected, Riemannian manifold. Then Mprin is simply-connected in M ∗

∗ Proof. Recall from Proposition 3.7 in [Mol88], that Σreg = Σreg/F is a Riemannian

orbifold since the leaves of (Σreg, F) are closed. Furthermore, from the fact that M is compact it follows that M is a complete Riemannian foliation. Therefore (M, F) is a full foliation. Since M is simply connected, by applying Lemma 4.26 it follows orb ∗ ∗ that the orbifold fundamental group π1 (Σreg) of Σreg is trivial. Therefore there ∗ are no codimension one strata in Σreg (see for example [Lan18]). Following the ∗ notation of Section 1.3 in [Dav11] if Σreg(1) denotes the complement of the strata ∗ of codimension at least 2, then Σreg(1) consists only of strata of codimension one ∗ and zero. Since the codimension one stratum is empty, then Σreg(1) corresponds exactly to the codimension zero stratum. From the fact that the codimension zero strata is the regular part of the orbifold (i.e. the manifold part), we conclude that ∗ ∗ ∗ Σreg(1) = Mprin. Taking x0 in the interior of Mprin, from Section 1.3 in [Dav11] the orb ∗ orbifold fundamental group π1 (Σreg, x0) is generated by taking π1(Mprin, x0) and adding the following generators:

∗ (i) For each component T of a codimension 2 stratum in the interior of Σreg,

choose a loop αT , starting at x0, which makes a small loop around T .

And the following relations: 4.4 weights of an a-foliation 71

n(T ) (ii) [αT ] , for some positive integer n(t).

∗ orb ∗ With this the group π1(Mprin, x0) is a subgroup of π1 (Σreg, x0) = 1. Thus we ∗ conclude that π1(Mprin, x0) = 1

Corollary 4.28. Consider a singular Riemannian foliation (M, F) with closed leaves on a compact simply-connected Riemannian manifold. Fix L0 and L principal ∗ ∗ leaves of F and consider two paths γ0 : I → Mprin and γ1 : I → Mprin, connecting ∗ ∗ L0 and L . Then the homeomorphism hγ0 is homotopic to hγ1 .

∗ Proof. From Lemma 4.27 we have that Mprin is simply connected. Therefore there is ∗ ∗ ∗ a homotopy from H : I → I → Mprin from γ0 to γ(1) fixing the end points L0 and L . ∗ This defines a continuous family of curves γs : I → Mprin, by setting γs(t) = H(t, s). x We define a homotopy Hf: L0 × I → L by setting Hf(x, s) = γs (1).

4.4 weights of an a-foliation

For a homogeneous A-foliation (M, F) of low codimension (i.e. one induced by an eﬀective torus action), Orlik and Raymond in [OR70], encoded the isotropy information of the orbits into weights of the orbit space M ∗. This approach was followed by Oh in [Oh83a], and Fintushel in [Fin77], to give equivariant classiﬁcations of homogeneous A-foliations by encoding the isotropy information as weights. In this section we extend the notion of weights to an arbitrary A-foliation (M, F) on a compact, simply-connected manifold M.

We start by ﬁxing a principal leaf L0. We consider any arbitrary point p ∈ M ⊥ and ﬁx it. Next we take v ∈ Sp , a normal vector to TpLp, such that q = expp(v) is ∗ contained in a principal leaf. From Theorem 3.12 there exists a path γ : I → Σreg ∗ ∗ q connecting q and L0. We consider the horizontal lift γ of γ, through q, and we 72 cross-sections and a-foliations

q set q0 = γ (1) ∈ L0. Recall from Section 3.3 that, in this setting, for some cover

Lp → Lp, we have a ﬁbration

Lv → Lq → Lp.(3.3.1)

From Theorem 4.15 (cf. [GGR15, Corollary B]) and Proposition 4.16, the principal n n−k k leaf Lq = T , Lp = T , and the leaf of the inﬁnitesimal foliation Lv = T , for some k 6 n. From the homotopy long exact sequences of the ﬁbration we get a short exact sequence

0 → π1(Lv, q) → π1(Lq, q) → π1(Lp, p) → 1.

∗ ∗ ∗ The path γ : I → Mprin connecting L0 to Lq induces a homeomorphism hγ : L0 → Lq. Via this homeomorphism, from the previous short sequence of homotopy groups of the ﬁbration, we obtain the following short exact sequence

0 → π1(Lv, q) → π1(L0, q0) → π1(Lp, p) → 1.

Since all spaces in this short exact sequence are tori, the short exact sequence becomes 0 → Zk → Zn → Zn−k → 0.

k Consider e1, ... , ek, generators of π1(Lv, q) = Z . They are mapped to elements n ap1, ... , apk in π1(L0, q0) = Z .

The deﬁnition of the integers ap1, ... , apk depends a priori on the choice of path ∗ ∗ γ joining L0 to Lq. The following lemma shows that in fact, they are independent of the choice of γ.

Lemma 4.29. The elements ap1, ... , apk ∈ π1(L0, q0) do not depend on the path γ : I → M ∗. 4.4 weights of an a-foliation 73

∗ ∗ Proof. If we choose any other path γ1 from L0 to Lq, then Corollary 4.28, shows that

the group isomorphisms induced by (hγ)∗ : π1(L0, q0) → π1(Lq, q) and

(hγ1 )∗ : π1(L0, q0) → π1(Lq, q) are equal. Therefore the set of integer vectors

ap1, ... , apk do not depend of the curve γ.

⊥ Next we prove that if we choose another vector w ∈ Sp such that expp(w) lies in

a principal leaf, then we recover the same integers ap1, ... , apk. ⊥ Lemma 4.30. The integers ap1, ... , apk do not depend on the choice of v ∈ Sp .

⊥ Proof. Take w ∈ Sp another vector with w 6= v, such that q1 = expp(w) lies ⊥ on a principal leaf Lq1 . Since (Sp , Fp) is a singular Riemannian foliation with ⊥ p closed, compact leaves, by Theorem 3.12, the space (Sp /F )prin is path-connected. ⊥ p ∗ ⊥ p ∗ Therefore there exists a path β : I → (Sp /F )prin from Lv ∈ Sp /F to Lw ∈ ⊥ p ⊥ p Sp /F . By taking horizontal lifts of β in (Sp , F )reg we obtain a homeomorphism 0 hβ : Lv → Lw. By setting q1 = expp(hβ(v)), the homeomorphism hβ induces an 0 isomorphism (hβ)∗ : π1(Lv, q) → π1(Lw, q1). From Corollary 4.28, this isomorphism is independent of the choice of β.

0 0 Let σ be a path in Lw from q1 to q1. This gives an isomorphism from π1(Lw, q1) 0 −1 onto π1(Lw, q1), given by mapping an element [δ] ∈ π1(Lw, q1) to [σ δσ]. Let α be 0 −1 −1 another path in Lw from q1 to q1. Consider the concatenation of paths σα δασ . −1 0 −1 −1 The path ασ is a loop based at q1. Thus we have a conjugation [σα ][δ][ασ ] 0 in π1(Lv, q1). Since we have an A-foliation, Lw is homeomorphic to a torus. Thus 0 −1 −1 π1(Lv, q1) is an Abelian group. Therefore the path σα δασ is homotopic to σ, relative to the end points. Thus α−1δα is homotopic to σ−1δσ. Therefore the 0 isomorphism from π1(Lw, q1) onto π1(Lw, q1), does not depend on the path σ. It

follows that we have a well deﬁned isomorphism from π1(Lv, q) to π1(Lw, q1).

Let hγ : Lq → L0 and hλ : Lq1 → L0, be homeomorphisms given by paths ∗ ∗ 0 γ : I → Mreg and λ: I → Mreg. Set x0 = hλ(q1), y0 = hλ(q1) and q0 = hγ(q)

(see Figure 4.2). Denote by i1 : Lv → Lq and i2 : Lw → Lq1 the inclusions, given by 74 cross-sections and a-foliations

the bundles (3.3.1), of the inﬁnitesimal leaves into the leaves Lq and Lq1 , respectively.

The homeomorphism hβ induces an isomorphism from (hγ ◦ i1)∗(π1(Lv, q) onto 0 (hλ ◦ i2)∗(π1(Lw, q1). The path σ : I → Lw gives a well deﬁned isomorphism from 0 (hγ ◦ i2)∗(π1(Lw, q1)) onto (hλ ◦ i2)∗(π1(Lw, q1)). Thus a generator of π1(Lv, q) in

π1(L0, q0) is mapped to a generator of π1(Lw, q1). From this we see that the integer

vectors ap1, ... , apk do not depend on v.

Figure 4.2.: Well deﬁned weights.

From the proof of the previous lemma, by using the fact that the fundamental

groups of Lp and L0 are Abelian, it follows that the deﬁnition of the integer vectors

ap1, ... , apk does not depend on the choice of basepoint p in Lp.

Lemma 4.31. The weights ap1, ... , apk of Lp do not depend on the choice of p ∈ Lp.

We recall from Proposition 4.18 that for a leaf Lp with non-trivial holonomy, Lp

is a Bieberbach space. The ﬁnite covering Lp → Lp implies the existence of the 4.4 weights of an a-foliation 75 following short exact sequence of groups, where the group H is the subgroup of the holonomy group ΓL ﬁxing v:

0 → π1(Lp, p) → π1(Lp, p) → H → 0.

We deﬁne the weights of the leaves of an A-foliation on a compact, simply- connected, manifold as follows. A principal leaf has no weight associated to it.

To an exceptional leaf Lp, we associate the collection {π1(Lp), H}. For a singular leaf Lp without holonomy we associate {ap1, ... , apk}. Finally, the weight of a singular leaf with holonomy Lp is the collection {ap1, ... , apk; π1(Lp, p), H}. With this information we can recover the homeomorphism type of a leaf, as well as its leaf type. Therefore we have encoded the leaf type information in the weights.

∗ ∗ We say two weighted leaf spaces, M1 and M2 , are isomorphic if there is a homeo- ∗ ∗ ∗ ∗ morphism ϕ: M1 → M2 sending the weights of M1 to the weights of M2 . The map ϕ is called an isomorphism between the weighted leaf spaces, or just simply an isomorphism between the leaf spaces. The following theorem, analogous to Theorem 4.13, shows the weighted space classiﬁes the topology of M as well as the foliation F.

Theorem D. If (M1, F1) and (M2, F2) are compact simply connected manifolds, with A-foliations, such that they have isomorphic weighted leaf spaces and admit ∗ cross-sections σi : Mi → Mi, then (M1, F1) is foliated homeomorphic to (M2, F2).

∗ ∗ ∗ Proof. Given a weighted isomorphism φ : M1 → M2 , between the leaf spaces we will deﬁne a foliated homeomorphism φ: (M1, F1) → (M2, F2). Fix x ∈ M1 and for ∗ ∗ the cross-section σ1 : M1 → M, set y = σ1(x ). The leaf Lx = Ly is homeomorphic k to R /Γ, where Γ is a Bieberbach group and 0 6 k ≤ dim(F). The Dirichlet domain D ⊂ Rk, of the action of Γ on Rk, is a convex fundamental domain (see for example [R¨o10, Theorem 2]). We may assume that a preimage of y corresponds to the center of the Dirichlet domain. Furthermore we may assume (via a translation) that in turn the center of the Dirichlet domain is the origin 0 ∈ Rk. Then there is 76 cross-sections and a-foliations

k a unique vector vx ∈ R connecting the origin to a preimage of x in the Dirichlet domain D.

∗ ∗ ∗ We set φ(y) = σ2(φ (y )). Since φ preserves the weights, then it preserves the k leaf type, and thus we have that Lφ(y) is homeomorphic to Ly = R /Γ. Last we set φ(x) as the point in the Dirichlet domain of φ(y) which corresponds to the vector

vx. In the same fashion we can construct a continuous foliated inverse map. Thus we have that φ is a foliated homeomorphism.

∗ Lets assume that the leaf spaces Mi admit a smooth structure (for example when they are homeomorphic to disks). In this case it may happen that the cross-sections ∗ σi : Mi → Mi are smooth. Furthermore if the leaves of (Mi, Fi) have a standard smooth structure then they are diﬀeomorphic to Rn/Γ. If this three hypothesis are met, i.e. the orbit space are smooth manifolds, the cross-sections are smooth, and

the leaves have a standard smooth structure the map φ: (M1, F1) → (M2, F2) is a foliated diﬀeomorphism.

Lemma 4.32. Let (M1, F1) and (M2, F) be compact, simply-connected manifolds, with A-foliations with standard diffeomorphism type, and isometric leaf spaces. If the ∗ ∗ leaf spaces M1 and M2 are homeomorphic to smooth manifolds, there is a weighted ∗ ∗ diffeomorphism f : M1 → M2, and the cross-sections σi : Mi → Mi are smooth with respect to these smooth structure, then the foliated homeomorphism of TheoremD is a foliated diffeomorphism.

Proof. This follows from the fact that the foliated homeomorphism defined in the proof of TheoremD, is defined by composition of the map f ∗ and the cross-sections, which by hypothesis are smooth. The fact that the leaves are diffeomorphic to Rn/Γ is used to show that once we have chosen our center of the Dirichlet domain y, the dependency of x ∈ L with respect to this center is smooth. 4.4 weights of an a-foliation 77

Let us consider (M, F) a compact, manifold with a singular Riemannian foliation admitting a cross-section σ : M ∗ → M. When M ∗ is homeomorphic to a smooth manifold (i.e. it admits a smooth structure), then the following lemma gives suﬃcient conditions to the existence of a smooth cross-section. Lemma 4.33. Let (M, F) be compact manifold with a singular Riemannian foliation. Assume that there is a cross-section σ : M ∗ → M, the leaf space M ∗ admits a smooth structure and the quotient map π : M → M ∗ is smooth with respect to this smooth structure. Then there exists a smooth cross-section σ : M ∗ → M.

Proof. We observe that by Theorem 3.3. in [Hir94], it follows that the space of smooth functions C∞(M ∗, M) is dense in the space of continuous functions C0(M ∗, M) with respect to the strong topology. Therefore there exists a smooth map h: M ∗ → M close to σ in C0(M ∗, M). Since the quotient map π : M → M ∗ is smooth, the map σ : M ∗ → M deﬁned as σ = h ◦ (π ◦ h)−1 is smooth. By construction the map σ is a cross-section for the map π : M → M ∗.

5 A-FOLIATIONS OF CODIMENSION 2

Using the frame work we developed in Chapter4, we will concentrate in this chapter on the study of A-foliations of codimension 2 on compact, simply-connected, Rie- mannian manifolds. In particular we will compare such foliations to homogeneous ones, and show that we can apply TheoremD. With this we will prove that any A-foliation of codimension 2 on a compact, simply-connected, Riemannian manifold is, up to foliated homeomorphism, a homogeneous foliation.

5.1 leaf space of a-foliations of codimension 2

We give a short review of how to prove that, for a compact, simply-connected manifold M with a singular A-foliation of codimension 2, the leaf space is a 2-disk. We begin by recalling that A-foliations of codimension 1 are homogeneous, and likewise regular A-foliations of codimension 2 are homogeneous, provided the manifold is closed and simply connected. Theorem 5.1 (Theorem E in [GGR15]). Let (M, F) be a simply-connected manifold equipped with regular a A-foliation of codimension 2. Then M = S3 and the foliation is given by a weighted Hopf action, or the following hold.

79 80 a-foliations of codimension 2

(i) The leaf space B = M/F is homeomorphic to a 2-disk, the interior of B is smooth, and the boundary ∂B consists of at least n totally geodesic segments meeting in an angle of π/2.

(ii) Let L0 be a generic leaf and L1 be a singular leaf. Then there is a submer- 1 sion L0 → L1, with ﬁber S if L1 belongs to a geodesic in ∂B, or with ﬁber 2 T , if L1 belongs to a vertex of ∂B.

Thus we will concentrate only on singular (i.e. where the dimensions of the leaves is not constant) A-foliations of codimension 2. Let (M, F) be a compact, simply-connected manifold with such a foliation. For p ∈ M we deﬁne the quotient codimension of the stratum Σp as:

codim(M, F) − codim(Σp, F).

Clearly, if (M, F) is a singular Riemannian foliation of codimension 2, then, for any p ∈ M, the quotient codimension of Σp is less than or equal to 2. Thus the following proposition establishes that codimension 2 singular Riemannian foliations are infinitesimally polar. Proposition 5.2 (Proposition 3.1 in [LT10]). Let (M, F) be a singular Riemannian foliation. Let x ∈ M be a point with stratum Σx of quotient codimension at most 2. Then F is infinitesimally polar at x. Corollary 5.3. Singular A-foliations of codimension 2 are infinitesimally polar.

With this information we see that the orbit space of a codimension 2 singular Riemannian foliation is an orbifold. For M simply-connected, applying the following theorems by Lytchak, we see that there are no exceptional leaves, and that the leaf space M ∗ has non-empty boundary. Furthermore the boundary corresponds to singular strata. Theorem 5.4 (Theorem 1.6 in [Lyt10]). Let (M, F) be a closed inﬁnitesimally polar singular Riemannian foliation on a complete manifold with quotient orbifold 5.1 leaf space of a-foliations of codimension 2 81

M ∗. Then all a singular leaves are contained in the boundary of ∂M ∗. If M is simply connected, then the quotient M ∗ has no boundary if and only if F is regular. Theorem 5.5 (Corollary 1.7 in [Lyt10]). Let (M, F) be a singular Riemannian foliation on a complete simply connected manifold, with quotient M ∗ of dimension 2. Then either the foliation is regular or there are no exceptional leaves.

As in the case of compact Lie group actions, the fundamental group of M surjects ∗ onto the fundamental group of the leaf space via π∗ : π1(M) → π1(M ) (see [Bre72, Chp. II, Thm. 6.2], [Bre72, Chp. II, Cor.6.3]). Therefore M ∗ is a simply-connected 2-orbifold with boundary. Thus it is homeomorphic to a 2-disk. The boundary ∂M ∗ is divided into k edges γi, and k vertexes Fi, labeled as pictured in Figure 5.1.

∗ The leaves that project under π : M → M to interior points of the arc γi, contained in ∂M ∗, which we call least singular leaves, are singular leaves of codimension 3 in M and thus are homeomorphic to T n−1. The leaves that project to the vertexes of ∂M ∗, called most singular leaves, are singular leaves homeomorphic to T n−2. For a point q in a singular leaf, we have by corollary 4.22 and [GGR15, Theorem D], ⊥ that the inﬁnitesimal foliation (Sq , Fq) at q is one of the homogeneous foliations (S2, S1) or (S3, T 2), induced by orthogonal actions.

The ﬁrst case occurs when the singular leaf projects to an interior point of an edge in M ∗, and the second case occurs when the singular leaf projects to a vertex. Since there are no exceptional leaves, the holonomy action is trivial. Thus for any point q ∈ M, we have Lq = Lq. Therefore, from Proposition 4.16 all the leaves of an A-foliation of codimension 2 are homeomorphic to tori. Furthermore for each edge in M ∗ we have the following type of ﬁbration:

S1 →T n → T n−1, (5.1.1) 82 a-foliations of codimension 2

F2 γ1 Fr γr T 2 ﬁber γ2 M ∗ Principal leaves

S1 ﬁber γ γ Fi i i+1 Fi+1

Figure 5.1.: Leaf space of A-foliation of codimension 2.

For each vertex in M ∗ we have the following two type of ﬁbration:

T 2 →T n → T n−2. (5.1.2)

In both cases the maps are smooth submersions.

5.2 weights of a-foliation of codimension 2

We know introduce the weights we developed in the preceding chapter for the special case of an A-foliation of codimension 2 on a compact, simply-connected manifold. We recall from the previous section that we have three types of leaves: the principal ones, the least singular ones, and the most singular ones.

n The weights of the least singular leaves, (ai1, ... , ain) ∈ Z correspond to the 1 1 n image of the generator αi ∈ π1(S ) under the inclusion π1(S ) → π1(T ).

Recall from Section 2.3 that for a homogeneous foliation given by a cohomogeneity 2 torus action, there are no exceptional orbits, and furthermore the singular orbits 5.3 top. classification of a-foliations of codim. 2 83

are tori Tn−1 and Tn−2. Furthermore the possible non-trivial isotropy groups are T1 and T2. Thus the weights for the leaves of codimension 3 in a homogeneous foliation given by a cohomogeneity two torus action, correspond to the following ﬁbrations: 1 n n−1 T = Gp → G = T → G/Gp = T .

Therefore the weights show how the isotropy subgroup T1 is immersed in the group Tn. From these observations it follows that the weights deﬁned for A-foliations coincide with the weights deﬁned by Oh for torus actions in [Oh83a].

5.3 topological classification of a-foliations of codimension 2

In this section we prove the following equivalence theorem for A-foliations of codimension 2 on compact, simply-connected (n + 2)-manifolds, which is one of our main results.

Theorem E. Let (M1, F1) and (M2, F2) be two compact, simply-connected smooth (n + 2)-manifolds, admitting singular A-foliations of codimension 2 and n > 2.

Then M1 is foliated homeomorphic to M2 if and only if the weighted leaf spaces

M1/F1 and M2/F2 are isomorphic.

Proof. We begin by observing that, for any A-foliation of codimension 2 on a compact, simply-connected, smooth, Riemannian manifold M the leaf space is a disk, which is a 2-dimensional CW-complex, with all interior points corresponding to principal leaves. Since the interior of the disk is contractible, by TheoremA, there is a cross-section deﬁned on the interior of the disk. Furthermore we note that the disk and its interior are homotopic equivalent, and thus we apply CorollaryC to get the 84 a-foliations of codimension 2

existence of a cross-section σ : M ∗ → M. Then applying TheoremD we get the desired conclusion.

Remark 5.6. For an A-foliation (M, F) of codimension 2 on a compact simply-

connected (n + 2)-manifold with n > 2, an other approach to obtaining a cross- section σ : M ∗ → M can be done using obstruction theory, and the procedure of Orlik and Raymond in [OR70]. Namely we can split the leaf space M ∗ into an open ∗ ∗ ∗ interior disk Y , and quadrilaterals D1, ... , Dr , as in Figure 5.2. Then we apply obstruction theory to show the existence of cross-section over Y ∗. We apply again ∗ obstruction theory to show that we can extend the given cross-section to D1. We ∗ then continue applying obstruction theory to extend the cross-section to each Di .

∗ ∗ Dr D1 ∗ D2 Y ∗ ∗ D3

Figure 5.2.: Decomposition of M ∗ in [OR70].

Moreover we remark that, for (M, F) a compact, simply-connected manifold with an A-foliation of codimension 2, the leaf space M ∗ is homeomorphic to a 2- disk. Thus the leaf space M ∗ admits a unique smooth structure. We will show that the hypothesis of Lemma 4.33 are satisﬁed in this case, and thus we may consider smooth cross-sections for A-foliations of codimension 2 . Lemma 5.7. Consider an A-foliation (M, F) of codimension 2 on a compact,

simply-connected manifold. Let p ∈ M be such that Lp is a least singular leaf (i.e.

Lp has codimension 3 in M). Then the following hold for the inﬁnitesimal foliation ⊥ p (Sp , F ) at p.

⊥ p (i) The quotient space Sp /F is homeomorphic to the closed interval [0, π], and thus it admits a unique smooth structure. 5.3 top. classification of a-foliations of codim. 2 85

⊥ ⊥ p (ii) The quotient map Sp → Sp /F is smooth.

⊥ p Proof. We note that (Sp , F ) is an A-foliation of codimension 1, with principal leaf 1 ⊥ p homeomorphic to S . It follows from Theorem D in [GGR15] that (Sp , F ) is the homogeneous foliation (S2, S1). Furthermore, from [GGZ39] and [Mos57] it follows that, any smooth action of S1 on S2 is equivalent (i.e. there exists an equivariant diﬀeomorphism) to the linear S1 action on S2. We describe this linear action. We consider, for S2, the following spherical coordinates:

(θ, ϕ) 7→ (sin θ cos ϕ, sin θ sin ϕ, cos θ),

with θ ∈ [0, π], ϕ ∈ [0, 2π], and parametrizing S1 by the angle ψ, the action of S1 on S2 is given by ψ(θ, ϕ) = (θ, ϕ + ψ).

Thus the quotient map S2 → S2/S1 is given by (θ, ϕ) 7→ θ (see Figure 5.3). This proves both claims.

Figure 5.3.: Quotient map of the homogeneous foliation (S2, S1)

Lemma 5.8. Consider an A-foliation (M, F) of codimension 2 on a compact,

simply-connected manifold. Let p ∈ M be such that Lp is a most singular leaf

(i.e. Lp has codimension 4 in M). Then the following hold for the inﬁnitesimal ⊥ p foliation (Sp , F ) at p. 86 a-foliations of codimension 2

⊥ p (i) The quotient space Sp /F is homeomorphic to the closed interval [0, π/2], and thus it admits a unique smooth structure.

⊥ ⊥ p (ii) The quotient map Sp → Sp /F is smooth.

⊥ p Proof. We note that (Sp , F ) is an A-foliation of codimension 1, with principal leaf 2 ⊥ p homeomorphic to T . It follows from Theorem D in [GGR15] that (Sp , F ) is the homogeneous foliation (S3, T2), given by the standard linear action.

We consider S3 as the unit sphere in C2 and we use the so-called Hopf coordinates for S3, given by

iθ1 iθ2 (θ1, θ2, η) 7→ (sin ηe , sin ηe , cos η),

with θ1 ∈ [0, 2π], θ2 ∈ [0, 2π], and η ∈ [0, π/2]. We parametrize the 2-torus T2 = S1 × S1 by the angles (α, β). With these coordinates the action of T2 on S3 is given by:

(α, β)(θ1, θ2, η) = (θ1 + α, θ2 + β, η).

3 3 2 Thus the quotient map S → S /T is given by (θ1, θ1, η) 7→ η. This proves both claims.

Proposition 5.9. For (M, F) a compact, simply-connected manifold with an A- foliation of codimension 2, the leaf space M ∗ admits a unique smooth structure. Furthermore there is a smooth cross-section σ : M ∗ → M with respect to this smooth structure.

Proof. Since, for a (singular) A-foliation (M, F) of codimension 2 on a simply- connected closed manifold the leaf space M ∗ is a 2-disk, it carries a unique smooth structure proving the first claim of the proposition. In this case in we get a smooth cross-section σ : M ∗ → M as follows. Let σ : M ∗ → M be a cross-section obtained ⊥ p from CorollaryC. Also we note that, for p ∈ M, the infinitesimal foliation (Sp , F ) is homogeneous (see Section 5.1). By Lemma 5.7 and Lemma 5.8, each infinitesimal 5.3 top. classification of a-foliations of codim. 2 87

⊥ ⊥ p foliation, the quotient map Sp → Sp /F is smooth. Since for any point p ∈ M, it has trivial holonomy group, a local neighborhood of p∗ is given by a cone over ⊥ p ∗ Sp /F . This implies that the quotient map π : M → M is smooth. Thus by applying Lemma 4.33 we obtain a smooth cross-section σ : M ∗ → M.

Before showing that A-foliations of codimension 2 on simply-connected manifolds are homogeneous we will state some facts about the weights.

From the proof of Theorem A in [GGR15], we are able to determine the number of diﬀerent bundles of the form (5.1.1) for an A-foliation of codimension 2 on a compact, simply-connected manifold. Theorem 5.10. Let (M, F) be a compact, simply-connected (n + 2)-manifold with an A-foliation of codimension 2, and L0 a regular leaf of dimension n. If the leaf ∗ space M has r-edges in the boundary, then r > n.

Proof. We first note that for A-foliations of codimension 2, a regular leaf is a prin- ∗ cipal leaf, and fix p0 ∈ L0. We consider M0 = Mreg, and B = B(M0 ) the Hae- ∗ fliger classifying space of M0 . Then from Theorem 4.23, Proposition 4.24, and Remark 4.25 we obtain the following long exact sequence:

· · · → π2(B, b0) → π1(L0, p0) → π1(M0, p0) → π1(B, b0) → 1.

By taking H to be the image of π2(B, b0) under the group morphism π2(B, b0) →

π1(L0, p0), we obtain the following short exact sequence:

0 → H → π1(L0, p0) → π1(M0, p0) → π1(B, b0) → 1.

Using the fact that for an A-foliation of codimension 2 on a compact, simply- connected manifold, the leaf space is a 2-disk, we conclude that H = 0. Consider the fibers of the fibrations given by the codimension 3 leafs. I.e. we consider the fibers of the fibrations of the from (5.1.1). Observe that by hypothesis, there are a 88 a-foliations of codimension 2

number r of these ﬁbrations. We consider their homotopy class in L0 and denote by

K the subgroup they generate in π1(L0, p0). It follows from the proof of Theorem A

in [GGR15] that π1(L0, p0) is generated by K and H. Furthermore K splits as an

Abelian group and a finite 2 step nilpotent 2-group. Since by Theorem 4.15 L0 is a torus, then we conclude that the finite 2 step nilpotent 2-group is trivial. Thus from this discussion it follows that there are at least n fibrations of the form (5.1.1).

Recalling Lemma 2.18, since the ﬁbers of the ﬁbrations of the form (5.1.1) generate a the fundamental group of a principal leaf, we deduce the following property n of the weights (ai1, ... , ain) ∈ Z , associated to the least singular leaves. Lemma 5.11. For an A foliation of codimension 2, the determinant of the weights

(a11, ... , a1n),(a21, ... , a2n), ... , (ak1, ... , akn) is ±1.

Now we are able to prove the main result for this chapter.

Theorem F. Let (M, F1) be closed, simply-connected (n + 2)-manifold with an A- foliation of codimension 2 and n > 2. Then there exist a closed, simply-connected

(n + 2)-manifold (N, F2) with a homogeneous A-foliation of codimension 2 (i.e. with

an eﬀective smooth torus action of cohomogeneity 2), such that (N, F2) is foliated

homeomorphic to (M, F1).

Proof. By Lemma 5.11, for an A-foliation of codimension 2 on a closed, simply-

connected (n + 2)-manifold M, the weights (ai1, ... , ain) are legal weights in the sense of Oh (see [Oh83a]). Thus by Theorem 2.22 there is a closed, simply-connected, (n + 2)-manifold N together with a Tn-action realizing the weights. By TheoremE, the manifolds M and N are foliated homeomorphic.

Remark 5.12. In the case of n = 2 or n = 3 the diﬀerentiable type of the torus is unique, so by Proposition 5.9 and Lemma 4.32, the homeomorphism in TheoremF will be a diﬀeomorphism. 6 SMOOTHSTRUCTUREOFLEAVESOF ANA-FOLIATION

As we have seen in Remark 5.12, for n = 2, 3, any A-foliation (M, F) of codimension 2 on a compact, simply-connected (n + 2)-manifold is, up to foliated diﬀeomorphism, homogeneous.

In this chapter, we prove that the same statement is true for n > 4. Namely we will show that for n > 4 an A-foliation (M, F) on a compact, simply-connected (n + 2)-manifold is, up to foliated diﬀeomorphism, homogeneous. In other words, we can strengthen the conclusions of TheoremF, to obtain the same result up to foliated diﬀeomorphism.

To achieve this, we need to study the diﬀerentiable structure of a principal leaf of (M, F), which, as proved in Section 4.2, is homeomorphic to a torus T k. Recall, as mentioned in the same section, for k > 5 there are examples of tori with exotic diﬀerentiable structure (see for example [HS70]). In the particular case of k = 4, for the torus T 4, to the best knowledge of the author, there is very few facts in the literature about its smooth structure (as it is with many other cases of four dimensional manifolds). In the present chapter we focus on studying the smooth structure of leaves of an A-foliation of codimension 2 on a compact, simply-connected manifolds.

89 90 smooth structure of leaves of an a-foliation

6.1 fibrations between leaves

We consider an A-foliation (M, F) on a compact, simply-connected manifold M.

Let p ∈ M be a ﬁxed point, and consider a nearby point q, such that the leaf Lq

has trivial holonomy. We recall that by Corollary 4.17 the leaf Lq is homeomorphic n to a torus T . We also observe that there is a ﬁnite covering Lp of the leaf Lp, with k Lp homeomorphic to a torus T . Now we consider the ﬁbration given by

ξ Lv → Lq → Lp.(3.3.1)

In particular from Proposition 4.16, ﬁbration (3.3.1) takes the form:

ξ T k → T m → T m−k. (6.1.1)

Remark 6.1. Furthermore if the leaf Lp has also trivial holonomy, recall that Lp is homeomorphic to T m−k.

Furthermore, we note that such fibrations come from an orthogonal metric projection (see Section 3.3), and thus this fibration is actually a smooth submersion. Since the total space of the fibration is a torus T n, it is compact. We restate Corollary 3.15 for completeness.

Corollary 3.15. Let W and N be smooth manifolds, with W compact. If f : W → N is a smooth surjective submersion, then f is a locally trivial ﬁbration.

Thus the fibration (6.1.1) is a locally trivial fibration, i.e. a fiber bundle. We collect this fact in the following result: Corollary 6.2. The submersions ξ : T m → T m−k are fiber bundles with fiber T k.

From now on we study the tori ﬁber bundles (6.1.1). 6.2 four dimensional torus. 91

6.2 four dimensional torus.

We begin by discussing the 4-dimensional case, i.e. when the total space of (6.1.1) is T 4. Let (M, F) be an A-foliation of codimension 2 on a compact-simply connected 6 manifold. In this case the least singular leaves are homeomorphic to T 3, and thus they admit a unique smooth structure. The least singular leaves are homeomorphic to T 2, which also admit a unique smooth structure. The only leaf type that may admit an exotic smooth structure is a principal leaf, which is homeomorphic to T 4.

∗ ∗ Consider x a point in M, such that Lx lies on a vertex of ∂M (i.e. Lx is a most singular leaf). Consider Lq a principal leaf with q close to p in M. For this particular setting. the ﬁbration (6.1.1) becomes:

T 2 → T 4 → T 2.

Even though, as mentioned before it is unstated in the literature, if the four torus admits a non-standard smooth structure, the total space of T 2 bundles over T 2 have been classiﬁed by Fukuhara and Sakamoto in [SF83]. Using this classiﬁcation, Ue showed in [Ue90] that a bundle of the form

T 2 → E → T 2, admits a geometric structure in the sense of Thurston (see for example [Sco83]). First, in [Ue90], the author proves that the total space E is classified among the geometric 4-manifolds, up to diffeomorphism, by π1(E). From this, it follows that it is enough to find a discrete faithful representation of π1(E) on a suitable geometry 92 smooth structure of leaves of an a-foliation

X. An explicit list of orientable T 2-bundles over T 2 can be found in [Gei92, Table 1]. From this table, it follows that for the case of the bundle:

T 2 → T 4 → T 2,

induced by the infinitesimal foliation (3.3.1), that the torus T 4 has an Euclidean geometry. This means that it is the quotient of Euclidean space, denoted by E4, by some group of finite isometries. This implies that a principal leaf of (M, F) is diffeomorphic to the standard torus. Since the other possible singular leaves (the least singular ones) are homeomorphic to T 3, and the 3-torus admits a unique smooth structure, it follows that for an A-foliation of codimension 2 on a compact, simply-connected 6-manifold all leaves are diffeomorphic to standard tori. Thus by Proposition 5.9, Lemma 4.32, Lemma 4.33, and TheoremF we prove that for a compact, simply-connected 6-manifold M, any A-foliation (M, F) of codimension 2 is homogeneous. Theorem 6.3. If (M, F) is a 6-dimensional, simply-connected, compact Rieman- nian manifold with an A-foliation of codimension 2, then the foliation is homogeneous.

6.3 higher dimensional torus.

In this section we show that, for n > 4, the smooth structure of the leaves of an A- foliation (M, F) of codimension 2 on a compact, simply-connected (n + 2)-manifold is the standard one. Recall that for (M, F) the least singular leaves are singular leaves of codimension 3 in M. The most singular leaves of (M, F) are singular leaves of codimension 4 in M. 6.3 higher dimensional torus. 93

We begin by recalling that, for the foliated manifold (M, F), the leaf space M ∗ ∗ is a disk. The boundary of M consists of a number r > n of edges, each edge denoted by γ, and a number r of vertexes. We label the edges and vertexes as in Figure 6.1.

∗ x1 ∗ ∗ x2 xr γ1 γr

γ2 M ∗

γ ∗ i γi+1 xi ∗ xi+1

Figure 6.1.: Labels in the leaf space of A-foliation of codimension 2.

We ﬁx p ∈ M, such that L∗ lies on the i-th edge, γ , in ∂M ∗ (i.e. L is a least i pi i pi n−1 singular leaf). This implies that Lpi is homeomorphic to T . Take qi ∈ M close enough to pi in M, such that the leaf Lqi is principal. Then Lqi is homeomorphic to T n. In this case the ﬁber bundle (6.1.1) takes the form:

1 n n−1 Si ,→ T → T . (6.3.1)

Remark 6.4. We note that we have exactly r of these bundles. One for each edge in ∂M ∗. The index i on the ﬁber is added to be able to distinguish the edge we are referring to.

With these bundles we will ﬁrst show a principal leaf (and thus all principal leaves) of an A-foliation of codimension 2 on a compact, simply-connected (n + 2)- manifold are the standard n-torus. We begin by observing that such bundles are orientable. Proposition 6.5. The bundle (6.3.1) is orientable. 94 smooth structure of leaves of an a-foliation

1 Proof. We can choose an arbitrary orientation for the fiber Si in local charts, to obtain a vector field, tangent to the circles in the total space. Since the n-torus is orientable, we can extend this vector field to a base, such that the transition maps have positive determinant in this base.

1 n Indeed if we choose on a local chart an orientation of the ﬁber Si ⊂ T , we can extend it to a basis of the tangent spaces of T n. Since T n is orientable we can do this construction in such a way that for two open trivial neighborhoods, the orientations of the ﬁbers are positive.

We combine the previous proposition with the following result, which tells us that the bundle (6.3.1) is principal. Theorem 6.6 (Proposition 6.15 in [Mor01]). Every oriented S1-bundle admits the structure of a principal S1-bundle. Corollary 6.7. The ﬁber bundle (6.3.1) is a principal S1-bundle.

We recall from TheoremF that an A-foliation (M, F) of codimension 2 on a compact, simply-connected (n + 2)-manifold is homogeneous, up to foliated home-

omorphism. Furthermore, from Theorem 5.10 we have r > n fiber bundles of the form (6.3.1). The circles, which are the fibers of these bundles, play the role of the isotropy circle subgroups for smooth (continuous) effective actions of T n on M. We n denote by (ai1, ai2, ... , ain) ∈ Z the weight associated to the leaf Lpi . We recall n that (ai1, ai2, ... , ain) defines how Si is embedded into the principal leaf T .

1 1 1 n Since the homotopy classes of the circles S1, S2, ··· , Sr ⊂ T generate the fun- n damental group π1(T ) of the principal leaf of (M, F), then by Lemma 5.11 there

exists a subcollection of labels {i1, i2, ... , in} such that the collection of weights

n o (ai11, ... , ai1n), ... , (ai11, ... , ai1n) , 6.3 higher dimensional torus. 95 have determinant ±1. We conclude from [Oh83a, Lemma 1.4], that the principal leaf T n of (M, F) is homeomorphic to the torus

1 1 Si1 × ... × Sin ,

∗ for the collection {i1, i2, ... , in} of distinct edge labels of ∂M .

From this observations, we can prove the following proposition: Proposition 6.8. There exists a free smooth Tn-action on the principal leaf T n of the foliation.

Proof. Consider the bundles S1 ,→ T n → T n−1, associated to the weights generating ij the fundamental group of the principal leaf T n. From Corollary 6.7 these bundles are principal. Thus for each i there is a free smooth action µ S1 × T n → T n. j ij : ij The image of this action is exactly the ﬁber S1 of the bundle (6.3.1). We now deﬁne ij the Tn-action µ: Tn × T n → T n on the principal leaf T n as

µ((ξ1, ... , ξn), p) = µi1 (ξ1, µi2 (ξ2, ··· , µin (ξn(p)) ··· )).

n The actions µij commute, since the principal leaf T is homeomorphic to the product S1 × · · · × S1 . Therefore µ gives a continuous action of the standard n-torus, Tn, i1 in on the principal leaf T n. Furthermore, the action µ is free and smooth since each of the transformations µij are free and smooth.

Corollary 6.9. For n ≥ 5, the principal leaf of an A-foliation (M, F) of codimension 2 on a compact, simply-connected manifold is diﬀeomorphic to the standard torus Tn.

Proof. Since T n = S1 × ... × S1 the action µ is transitive, and therefore T n is the i1 in standard torus Tn. 96 smooth structure of leaves of an a-foliation

We end this section by proving that the singular leaves of an A-foliation of codimension 2 on a compact, simply-connected manifold are also diﬀeomorphic to standard tori. Corollary 6.10. The least singular leaf of an A-foliation (M, F) of codimension 2 on a compact, simply-connected (n + 2)-manifold M is diﬀeomorphic to the standard torus.

Proof. For the least singular leaf Lpi the claim follows from the fact that the ﬁber 1 bundle (6.3.1) is an Si -principal bundle, combined with the fact that the total n space is the standard torus T . Thus the least singular leaf Lpi is diﬀeomorphic to n 1 n−1 T /Si = T , i.e. the standard (n − 1)-dimensional torus.

We recall that, if x is a point in M, F , such that L∗ is a vertex in M ∗, then i ( ) xi n−1 Lxi is a most singular leaf of F, and it is homeomorphic to T . Furthermore we

can choose pi close enough to xi in M, such that Lpi is a least singular leaf. We

point out that the leaf Lpi has trivial holonomy. Thus for the leaves Lpi and Lxi ﬁbration (3.3.1) is a ﬁber bundle of the form:

S1 ,→ T n−1 → T n−2. (6.3.2)

By following the same arguments as in the proof of Proposition 6.5 we prove that this bundle is orientable. Applying Theorem 6.6 we conclude that the ﬁbration (6.3.2) is a principal S1-bundle. With theses remarks we can prove the following proposition: Proposition 6.11. The most singular leaf of an A-foliation (M, F) of codimension 2 on a compact, simply-connected (n + 2)-manifold M is diﬀeomorphic to the standard torus. 6.3 higher dimensional torus. 97

Proof. Since we have a smooth principal S1-bundle:

1 S ,→ Lpi → Lxi ,(6.3.2)

1 we conclude that Lxi is diffeomorphic to Lpi /S . From Corollary 6.10, we have that n−1 the least singular leaf Lpi is diffeomorphic to T . Thus the most singular leaf is diffeomorphic to Tn−1/S1 = Tn−2.

Thus all leaves in an A-foliation (M, F) of codimension 2 on a compact simply- connected (n + 2)-manifold M, are diﬀeomorphic to standard tori. Following the proof of TheoremF, together with Lemma 4.32, and Remark 5.9 we get a proof of the main theorem of the present work: Theorem G. Every A-foliation of codimension 2 on a compact, simply-connected Riemannian manifold is homogeneous.

APPENDIX

A LINEARIZEDFLOWS

In this appendix we prove that for a singular Riemannian foliation (M, F), given ⊥ p a closed leaf L and a point p ∈ L, the map ρ: π1(L, p) → O(Sp , F ) deﬁned in Section 3.3 is well deﬁned. To do this, we need to understand the correspondence given in Theorem 3.8 between a path γ : [0, 1] → L, starting at p, and a foliated map G: νpL → νL. We recall the needed concepts and results from Section 3.2 in [MR18], and the notes [Rad17].

1 linearized vector fields

We consider a complete Riemannian manifold M, and L a closed submanifold of M. Let X be a vector field on M, which is tangent to L when restricted to L. Recall that there exists an open neighborhood W ⊂ νL of the zero-section of the normal bundle νL → L, and an open neighborhood U ⊂ M of L, such that normal exponential map exp⊥ : W → U is a diffeomorphism. We will consider the preimage ⊥ −1 (exp )∗ (X) of X, which we will also denote by X. Observe that this preimage is also a smooth vector field on W . Given λ > 0 we consider the transformation rλ : νL → νL given by taking a normal vector field V to λV . If λ is small enough, the image of the restriction of rλ to W lies in W again, i.e. rλ(W ) ⊂ W .

101 102 linearized flows

Given a smooth vector field X on W ⊂ νL, we define the linearization of X around L, denoted by X`, as the vector field obtained as the limit

` −1 X = lim (rλ)∗ (X ◦ rλ). λ→0

If X` = X, we say that X is a linearized vector field. By the following proposition ` ` shows that X is well defined and it is invariant under rescalings, i.e. (rλ)∗(X ) = X`. Proposition A.1 (Linearization of vector fields, Proposition 13 in [MR18]). Let X be a smooth vector field on W . Then its linearization X` is a well-defined, smooth vector field defined on the whole of νL, which is invariant under rescalings.

Now we consider the case where L is a closed leaf of a singular Riemannian foliation (M, F). Recall, from Section 3.2, that there exists a singular foliation, which we denote in this appendix by νF, on νL, which is scaling invariant. Its ⊥ leaves are the preimages of F|U given by the map exp . The next proposition shows that flows of linearized vectors preserve this foliation. Proposition A.2 (Linear flows, Proposition 14 in [MR18]). Let (M, F) be a singular Riemannian foliation, L a closed leaf, and X a vector field tangent to the leaves of F. Then for any t ∈ R, the linearization X` around L and its flow Φt : νL → νL satisfy:

(i) X` is a tangent to the leaves of the singular foliation (νL, νF), and Φt preserves the leaves of (νL, νF).

t (ii) For any p ∈ L, the restriction of Φ to νpL is a linear orthogonal transforma-

tion from νpL to νΦt(p)L.

We recall the following property of singular Riemannian foliations, the so-called equifocallity, as stated in [MR18]. 1 linearized vector fields 103

Theorem A.3 (Equifocality, Proposition 5 in [MR18]). Let F be a singular Rie- mannian foliation of a Riemannian manifold M, and let L be a leaf. If v, w ∈ νL are two normal vectors (at possibly diﬀerent points) such that exp⊥(tv) and exp⊥(tw) belong to the same leaf for all small t > 0, then they belong to the same leaf for all t ∈ R for which exp⊥(tv) and exp⊥(tw) are deﬁned.

Proof. See Proposition 4.3 in [LT10], and Theorem 2.9 in [Ale10].

With Proposition A.2 and Theorem A.3 we are able to give a proof of Theorem 3.8, as done in [MR18].

Theorem 3.8. Let L be a closed leaf of a singular Riemannian foliation (M, F) , and γ : [0, 1] → L a piece-wise smooth curve with γ(0) = p. Then there is a map

G: [0, 1] × νpL → νL such that:

(i) G(t, v) ∈ νγ(t)L for every (t, v) ∈ [0, 1] × νpL.

(ii) For every t ∈ [0, 1], the restriction G: {t} × νpL → νγ(t)L is a linear

isometry preserving the leaves of νL.

(iii) For every s ∈ R the map expγ(t)(sG(t, v)) belongs to the same leaf as the

point expp(sv).

Proof. First we consider a partition 0 = t0 < t1 < ··· < tN = 1 of [0, 1], such that the restriction γi = γ : [ti−1, ti] → L is an embedding for 1 6 i 6 N. Thus for all i, the curves γi are integral curves of some smooth vector field Xi on L. We extend each vector field Xi to M, and obtain a vector field Xˆ i on M. We then use the

Riemannian metric to consider the component of Xˆ i tangent to the leaves of F. In this way we obtain a vector ﬁeld, which we denote by Xi, which is an extension of the original Xi. Observe that by construction Xi is tangent to the leaves. Next we consider a neighborhood W of the zero section in νL, and U ⊂ M a neighborhood 104 linearized flows

of L such that exp⊥ : W → U is a diffeomorphism. We identify each vector field ⊥ Xi with a vector field Xi on W via the inverse of (exp )∗. We now consider the ` t linearizations Xi around L of the vector fields Xi, and we denote by Φi the flow ` of each Xi . Given a normal vector v ∈ νpL, by Proposition A.2, on [0, t1] we

t1 have a linear orthogonal transformation Φ1 (v) ∈ νγ(t1). We may apply the same

construction on [t1, t2] for v1 ∈ νγ(t1)L, and so on for the rest of the partition of [0, 1].

In this way we can deﬁne a map G: [0, 1] × νpL → νL as follows: for t ∈ [tj−1, j]

and v ∈ νpL we deﬁne G as

t−tj−1 tj−1−tj−2 t2−t1 t1 G(t, v) = Φj ◦ Φj−1 ◦ · · · ◦ Φ2 ◦ Φ1 (v).

Parts (i) and (ii) follow from Proposition A.2. Part (iii) follows from Theorem A.3.

Consider two curves, γ0 and γ1, in a closed leaf L, with γ0(0) = p = γ1(0)

and γ0(1) = q = γ1(1). Let Gi : νp → νq be the linear transformation given

by Theorem 3.8, associated to γi. We end this appendix by showing that, if the −1 two paths γ0 and γ1 are homotopic relative to the endpoints, then (G1) ◦ G0

is homotopic to the identity map of νpL. We follow the proof of Lemma 2.36 in [Rad17].

Proposition A.4. Let γ0 and γ1 be two curves in a closed leaf L which are homo-

topic relative to the end points, with γ0(0) = p = γ1(0), and γ0(1) = q = γ1(1). −1 Then (G1) ◦ G0 : νpL → νp is homotopic to the identity map. Furthermore it takes every leaf of the inﬁnitesimal foliation F p to itself.

Proof. Let H : [0, 1] × I → L be the homotopy between γ0 and γ1. By applying Whitney’s Approximation Theorem (see for example Theorem 9.27 in [Lee13]), we can assume that H is a smooth map. For s ∈ I ﬁxed we consider the smooth

curve γs(t) = H(t, s). From the compactness of [0, 1] × I we can ﬁnd a partition

0 = t0 < t1 < ··· tN = 1 of [0, 1] such that for any s ∈ I the curves γs restricted to 1 linearized vector fields 105

0 [ti−1, ti] is an embedding. By extending the vector field γs(t) for t ∈ [ti−1, ti] to L, we obtain smooth vector fields Vsi on L. Since the family of curves γs varies continuously with respect to s by construction, for each 1 6 i 6 N the family of vector fields Vsi varies smoothly with respect to s. This implies that when we consider for each γs the map Gs : νpL → νqL given by Theorem 3.8, then Gs varies continuously with −1 respect to s (see Proof of Theorem 3.8). Defining K(v, s) = (Gs) (G0(v)) we obtain a homotopy K : νpL × I → νpL, between the identity Id: νpL → νpL and −1 (G1) ◦ G0 : νpL → νpL. For v ∈ νpL fixed, we have, from Theorem 3.8(iii), that −1 expp((Gs) (G0(v))) lies in the same leaf of F as expp(v). Since K(v, s) defines a −1 −1 path between v and (G1) (G0(v)), we have that (G1) (G0(v)) lies in the same p −1 leaf Lv of F as v. Thus (G1) (G0(Lv)) ⊂ Lv.

⊥ p Recall from Section 3.3, that the group O(Sp , F ) consists of all the foliated isometries of the infinitesimal foliation at p, and the subgroup O(F p) consists of all the foliated isometries which leave invariant the leaves of F p. The last part of −1 p Proposition A.4 states that (G1) ◦ G0 is an element of O(F ). Corollary A.5. For a point p in a closed leaf L of a singular Riemannian foliation ⊥ p p (M, F), the map ρ: π1(Lp, p) → O(Sp , F )/O(F ),defined in Section 3.3 is well defined.

Proof. We recall how the map ρ is deﬁned. Given a loop γ0, we consider G0 : νpL →

νpL the linear foliated transformation given by Theorem 3.8, and set ρ[γ0] = [G0] ∈ ⊥ p p O(Sp , F )/O(F ). From Proposition A.4 if γ1 is a loop homotopic to γ0, then we −1 p ⊥ p p have (G1) ◦ G0 ∈ O(F ). Therefore [G0] = [G1] in O(Sp , F )/O(F ).

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